%%
%% This is the paper for the Marinov Memorial Volume
%%
%% "CLASSICAL INTEGRABLE SYSTEMS:SELECTED TOPICS"
%% by A.M.Perelomov
%%
%% Email: perelomo@dftuz.unizar.es
%% Address:
%% Prof. A.M.Perelomov
%% Depart. de Fisica
%% Fac. de Ciencias
%% Univ. de Oviedo
%% Av. Calvo Sotelo 18
%% 33007 OVIEDO
%% SPAIN
%%

\documentstyle[amssymb,sprocl,twoside]{article}
%% Macroses below are optional, you can introduce your own ones.

%% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

%% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}

%% Some other macros used in the sample text
\def\b{\hfil\break}
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacemt due to no font
\def\tr{\,{\rm tr}\,}
\def\diag{\,{\rm diag}\,}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% macros for boxes
\catcode`@=11
\def\alloc@#1#2#3#4#5{\global\advance\count1#1by\@ne
  \ch@ck#1#4#2% make sure there's still room
  \allocationnumber=\count1#1%
  \global#3#5=\allocationnumber
  \writelog{#5}{#2}}
\def\writelog#1#2{}
\newif\ifopened  \openedfalse
\def\openup{\openedtrue\afterassignment\@penup\dimen@=}
\catcode`@=12 % at signs are no longer letters
\def\boxdisp#1{\vcenter{\boxgen{\hbox{$\displaystyle #1$}}{10pt}}}
\def\boxtext#1{\boxgen{#1}{3pt}}
\def\boxeq#1{\boxgen{\hbox{$#1$}}{3pt}}
\def\boxgen#1#2{\hbox{\vrule\vbox{\hrule\kern#2\vtop{
   \hbox{\kern#2#1\kern#2}\kern#2\hrule}}\vrule}}
\def\gboxit#1{\ifopened\boxdisp{#1}\else\ifmmode
   \ifinner\boxeq{#1}\else\boxdisp{#1}\fi
  \else\boxtext{#1}\fi\fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\citebk#1{\hspace{0.9mm}\raisebox{-1.85mm}[0mm][0mm]
  {\Large\cite{#1}}\hspace{-0.1mm}}

\def\citebkcap#1{\hspace{0.8mm}\raisebox{-1.5mm}[0mm][0mm]
  {\large\cite{#1}}\hspace{-0.2mm}}

\pagestyle{myheadings}  %% {Authors}{Title}
\markboth{\small \em \quad A. M. Perelomov\hfill}{\hfill\small \em Classical
integrable systems\quad}

\thispagestyle{plain}
\setcounter{page}{267}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\sloppy
\medskip
\rightline{\sl In memory of M.S. Marinov}
\medskip

\title{CLASSICAL INTEGRABLE SYSTEMS: SELECTED TOPICS$^{\,\dagger}$}
\footnotetext{$^\dagger$ This paper based partially on Lausanne 
lectures.\cite{[Pe 1995]}}

\author{A. M. PERELOMOV}
\address{Institute of Theoretical and Experimental Physics,\\
117259 Moscow, Russia}
\maketitle 

\abstracts{ 
In this paper two topics are considered:
\\[1mm]
$\bullet$ Poisson structures and Hamiltonian systems.\\[1mm]
$\bullet$ Explicit integration of equations of motion for the
Calogero--Sutherland and\\ $~~~$Toda systems.\\[1mm]
It must be said that a part of the material is well known
to specialists in this field for a long time
(see for example the book\,\cite{[Pe 1990]}). Even so, it may not always 
be easily
accessible to nonspecialists. Within the limited space it was not possible to
give a completely systematic exposition of these topics. Rather I
tried to illustrate the main ideas on simple examples and in an as
elementary and explicit way as possible. The examples considered here are
therefore restricted to the systems related to the algebras
$A_{n}\sim su_{n+1}$. For the case of arbitrary simple Lie
algebra, we have only mentioned the results at best. Due to
insufficient space many important results are omitted here.
Nevertheless I hope that this paper may be useful and may serve
as a guide, so that the interested reader may proceed to read
original articles. }

\vspace{0.4cm}

\tableofcontents

\newpage
%%%%%%%%
\setcounter{footnote}{0}

\section{Poisson Structures and Hamiltonian Systems}
\subsection{Basic definitions}\label{subsec:basic}


In order to motivate the basic concepts of Hamiltonian systems,
let us begin with the simplest example.

Suppose a particle with mass $m$ moves in a field $U(q)$, where
$q=(q^{1},\ldots ,q^{n})$ is a vector of $n$-dimensional space.
The motion of the particle is described by Newton's equations
\begin{equation}
m\ddot {q}^{j}= -\frac{\partial U}{\partial q^{j}},
\end{equation}
where ``dot'' means the differentiation in time.

Let us introduce as usual the momentum vector
\begin{equation}
p = (p_{1},\ldots ,p_{n}),\quad p_{j} = m\dot {q}^{j},
\end{equation}
and the Hamilton's  function or Hamiltonian
\begin{equation}
H = \frac{1}{2m}\,p^{2}+U(q),\quad p^{2} = p_{j}p_{j} \equiv \sum _{j=1}^{n}
p_{j}p_{j}.
\end{equation}
The Newton equations may then be rewritten in the form of the
Hamilton equations
\begin{equation}
\dot {q}^{j} = \frac{\partial H}{\partial p_{j}},\quad \dot {p}_{j} =
-\frac{\partial H}{\partial q^{j}},
\end{equation}
and, in general, the dependence of $H$ on $p$ and $q$ may be
arbitrary. Equations (4) may be rewritten as one equation.\,\, To
do this,\,\, let us unify the vectors $p$ and $q$ into one
$2n$-dimensional vector $x=(p,q)$, the quantities $\left(\frac{\partial
H}{\partial p_{j}},\,\frac{\partial H}{\partial q^{k}}\right)$ into the
$2n$-dimensional vector $\nabla H$, and let us introduce the
matrix ${\rm J}$ of order $2n$:
\begin{displaymath}
\mbox{J} = \left(\begin{array}{cc}
0&{\rm I}\\
{\rm -I}&0
\end{array} \right), \end{displaymath}
where ${\rm I}$ is the unit matrix of order $n$. Then the Hamilton
equations (4) may be written as\,\footnote{\,Such a form of the Hamilton 
equations
was used first, probably, by Lagrange\,\cite{[La 1808],[La 1809]} for 
the description
of the variation of elements of planets perturbed by the action of other
planets.}
\begin{equation}
\dot x = J\cdot \nabla \,H(x)\quad ({\rm or}\,\,J\cdot \dot {x} = -\nabla H),
\end{equation}
i.e.~as the equation in the phase space
$M=\{x\},\,\,x=(x^{1},\ldots ,x^{2n})$,
which in our case is the $2n$-dimensional Euclidean space.

Using the matrix $J$ we may define the Poisson
bracket\,\footnote{\,Such brackets, for which we use the name ``standard", were
introduced by Poisson.\cite{[Po 1809]}} in the space ${\cal F}(M)$ of the smooth
functions on $M$, for which we use the name {\em standard} or {\em canonical
Poisson bracket}:
\begin{equation}
\{ f(x),\,g(x) \} = (\nabla f,\,\,J\,\nabla g) = J^{jk}\,\partial _{j}f\,
\partial _{k}g \equiv \sum _{j=1}^{n}\,\left(\frac{\partial f}{\partial
p_{j}}\,\frac{\partial g}{\partial q^{j}} - \frac{\partial f}{\partial q^{j}}
\,\frac{\partial g}{\partial p_{j}}\right).
\end{equation}

It is easy to check that the standard Poisson bracket satisfies the following
three conditions:
\begin{description}
\item[(i)] antisymmetry
\be \{ f,g\} = -\{ g,f\}; \ee

\item[(ii)] the Leibniz rule
\be \{ f,gh\} = g\,\{ f,h\}+h\,\{ f,g\}; \ee

\item[(iii)] the Jacobi identity
\be \{ \,f\,\{g,h\}\,\} + \{\,g\,\{ h,f\}\,\} + \{h\,\{ f,g\}\,\} = 0. \ee
\end{description}

Using this Poisson bracket we may rewrite the Hamilton equations in the form:
\be \dot x = \{ H,x \}.\ee
It follows from the Jacobi identity (9) that if the functions $H_{1}(x)$
and $H_{2}(x)$ are both integrals of motion, i.e.~$\dot H_{1}(x)=0$ and
$\dot H_{2}(x)=0$ due to the equation of motion (10), then the quantity
\be H_{3}(x) = \{\,H_{1}(x),\,H_{2}(x)\,\}\ee
will be also an integral of motion.

Now we are ready to give the general (or abstract) definition of the
Hamiltonian system.

Let $M$ be a manifold and ${\cal F}\,(M)$ be the space of smooth functions
on $M$.
\newline \newline
{\bf Definition 1}. {\em The Poisson bracket on $M$ is a bilinear operation
on ${\cal F}\,(M)$ satisfying the conditions (7), (8) and (9).}

The conditions (7) and (9) are none other than the axioms of the Lie algebra.
So the space ${\cal F}\,(M)$ equipped with the Poisson bracket $\{\,\cdot ,
\cdot \,\}$ becomes the Lie algebra (in the general case, an
infinite-dimensional one).

As for the condition (8), it follows from the fact that the Poisson bracket
is defined by the Poisson bracket for the local coordinates
\be \{ \,x^{j},x^{k}\,\} = \omega ^{jk}\,(x),\ee
i.e.~by the antisymmetric tensor $\omega ^{jk}(x)$.

In fact, we have
\be \{ \,f(x),\,g(x)\,\} = \omega ^{jk}\,(x)\,\partial _{j}\,f(x)\,
\partial _{k}\,g(x).\ee

Let us note that for this Poisson bracket the Leibniz rule (8) is
satisfied automatically while the two conditions (7) and (9) are
equivalent to the conditions \be \omega ^{jk}\,(x) = -\omega
^{kj}\,(x),\ee and \be \omega ^{jk}\,\partial _{k}\,\omega ^{lm} +
\omega ^{lk}\,\partial _{k}\,\omega ^{mj} + \omega ^{mk}\,\partial
_{k}\,\omega ^{jl} = 0.\ee

We say that we have the Poisson structure on the manifold $M$, if
the Poisson bracket on ${\cal F}\,(M)$ satisfies the conditions
(7), (8) and (9). Such Poisson structure may be defined by the
bi-vector \be \Lambda  = \omega ^{jk}\,\partial _{j}\,\wedge
\,\partial _{k}, \ee where $\partial _{j}\,\wedge \,\partial _{k}$
means the wedge product of the vector fields $\partial _{j}$ and
$\partial _{k}$.

Now we can define the notation of the Hamiltonian system.
\newline \newline
\noindent{\bf Definition 2}. {\em The dynamical system
\be \dot x = X \ee
is called a Hamiltonian system, if the vector field $X$ is a
Hamiltonian vector field, i.e.~if
\be X = X_{H} = \omega ^{jk}(x)\,(\partial _{j}H)\,\partial _{k}, \ee
or in another way,
\be \dot x = \{\,H,x\,\}.\ee}

So a general Hamiltonian system is the triple
$$ M,\,\,\{\,,\,\},\,\,H(x), $$
where $M$ is a manifold, $\{\,,\,\}$ is the Poisson structure on $M$ and
$H(x)$ is a real function on $M$.

Let us emphasize that the tensor $\omega ^{jk}(x)$ appearing in
(16) need not be non-degenerate. In particular, the dimension of
$M$ may be odd. The simplest case of such a situation is the
three-dimensional case.

%\newpage
\subsection{Examples of Poisson structures}
\setcounter{equation}{0}

Following the paper\,\cite{[GMP 1993]} (see also 
Ref.\,\protect\citebk{[Pe 1990]}) let us discuss
typical examples of the Poisson bracket.
\newline
\begin{itemize}
\item{\bf Example 1.}
The standard or canonical Poisson bracket on a $2n$-dimensional vector space
may be written in canonical coordinates $({\bf q},{\bf p})$ in the form
\be \{ f,g\} =\sum_{j=1}^{n} \left({\partial f\over \partial p_j}\,
{\partial g\over \partial q_j} - {\partial f\over \partial q_j}\,
{\partial g\over \partial p_j}\right).\ee

\item{\bf Example 2.}
If a tensor $\omega ^{jk}(x)$ does not depend on $x$, by a linear change of
the variables this case can be reduced to the bracket as above,
but with $({\bf q},{\bf p})$ being only part of the coordinates.
The canonical Poisson bracket will be produced if these are all the
coordinates, which is possible only in an even dimension.

\item{\bf Example 3.}
The next important case is that of the linear dependence of
$\omega ^{jk}(x)$ on $x$ (here let us assume for simplicity that
$M$ is a linear space): \be \omega ^{jk}(x) = C_{l}^{jk}x^{l}.\ee
Since \be \{x^{j}, x^{k}\} = C^{jk}_{l}x^{l},\ee $C_{l}^{jk}$ are
structure constants of some Lie algebra. Conversely, having the
Lie algebra $({\cal G},[\cdot ,\cdot ])$ with a basis $e^l,$ we
can regard this basis as the coordinate $x^{l}$ (in fact linear)
system for the dual space ${\cal G}^*$, and consider the unique
Poisson bracket $\{ \cdot ,\cdot \}$ on ${\cal G}^*$ for which $\{
x^i,x^j\} =[e^i,e^j].$ The Jacobi identity for the bracket in
${\cal G}$ implies that this bracket on ${\cal F}({\cal G}^*)$
given by \be \{f(x), g(x)\} = C_{l}^{jk}x^{l}\partial
_{j}f\,\partial _{k}g\ee is really the Poisson bracket. This is
usually called the {\em Kirillov-Kostant-Souriau bracket} (or
simply KKS-bracket)\,\footnote{\,Note, however, that this bracket was
known to S.~Lie.\cite{[Lie 1874/75]} } and can also be written formally as
\be \{f(x), g(x)\} = \left(x, \left[ {\partial f\over
\partial x},\, {\partial g\over \partial x}\right] \right).\ee
In this case the Poisson bracket of two polynomial functions on
${\cal G}^*$ is again a polynomial function, so that the space
${\cal P}({\cal G}^*)$ of all polynomials on ${\cal G}^*$ is the
Lie subalgebra.

\item{\bf Example 4.}
Having associated our bracket with a tensor field, it is clear that
we can perform a coordinate transformation and generate ``new" brackets,
if the transformation is not canonical.

Let $M$ be the dual space of the Lie algebra of the group
$SO(3),\,\, M= \{ x : x = (x_1, x_2, x_3)\}$. In this case, the
KKS-bracket is given by \be \{x_{j}, x_{k}\} =\varepsilon
_{jkl}x_{l},\quad j, k, l = 1, 2, 3,\ee where $\varepsilon _{jkl}$
is a totally skew-symmetric tensor, $\varepsilon _{123} = 1.$ The
Poisson structure is degenerate, and on the orbits ${\cal O}_{r} =
\{x : |x|^2 = x_1^2 + x_2^2 + x_3^2 = r^2\}$ of the co-adjoint
representation it can be expressed in spherical coordinates as \be
\{f(\theta , \varphi),\,g(\theta ,\varphi)\} = {1\over {r\sin
\theta }}\, \left({\partial f\over
\partial \theta }\,{\partial g\over \partial \varphi }
 - {\partial f\over \partial \varphi }\,{\partial g\over \partial \theta }
 \right).\ee
Here $x_1 = r\sin {\theta } \cos {\varphi },\,\,x_2 = r\sin {\theta }
\sin \varphi,\,\,x_3 = r\cos \theta $.
The dynamics in $M$ is given by the equation
\be \dot x = \left[x, {\partial H\over \partial x}\right],\ee
where $[\,,\,]$ is the vector product.
We notice that for a quadratic Hamiltonian
\be H = {1\over 2}\sum {a_j x_{j}^{2}}\ee
these equations turn into Euler's equations describing the motion of a rigid
body about a fixed point.

\item{\bf Example 5.}
Consider a Poisson structure $\Lambda =\omega ^{jk}\partial _j\wedge
\partial_k$ with constant coefficients. Using the coordinate
transformation $\xi _i=e^{x_i}$ we get the quadratic bracket $\{ \xi _j,
\xi _k \} =\omega ^{jk}\xi _j\xi _k$ corresponding to the Poisson
structure
\be \Lambda =\omega ^{jk}\xi _j\xi _k{\partial \over \partial
\xi _j}\wedge {\partial \over \partial \xi _k}.\ee
\end{itemize}

\subsection{The three-dimensional case}
\setcounter{equation}{0}

In this section, following Ref.\,\protect\citebk{[GMP 1993]}, we give the description of
the general Poisson structure for the case of a three-dimensional
manifold.

Let us consider for simplicity the case of three-dimensional
Euclidean space $M = \{\bf R\}^{3}$ with standard coordinates
$(x^{1},x^{2},x^{3})$. Then the Poisson structure is defined by
three Poisson brackets \be \{ x^{2},x^{3}\} = A_{1}({\bf
x}),\,\,\{x^{3}, x^{1}\} = A_{2}({\bf x}),\, \,\{x^{1}, x^{2}\} =
A_{3}({\bf x})\ee or by the three-dimensional vector \be {\bf A} =
(A_{1}({\bf x}),\,A_{2}({\bf x}),\,A_{3}({\bf x})).\ee We have
here just one Jacobi identity, which has the form \be {\bf
A}\,{\rm rot}\,{\bf A} = 0.\ee It is easy to check that the
vector-field ${\bf A}({\bf x})=u({\bf x})\, \nabla \varphi ({\bf
x})$ is the solution of this equation.

Let us show that this solution is in fact the general local
solution of this equation. Let us introduce the one-form \be
\alpha =A_{j}({\bf x})\,{\rm d}x^{j}.\ee Then equation (3) takes
the form \be \alpha \land {\rm d}\alpha =0,\ee where $\land $ is
the exterior product. However, it is known that in this case the
one-form $\alpha $ admits an integrating factor. So we can write
$\alpha =u\,{\rm d}\varphi $ for some smooth functions $u, \varphi
$ defined at least on an open and dense subset of the space ${\bf
R}^{3}$.  Thus we get the following theorem.
\newline \newline
\noindent{\bf Theorem 1.} {\em Every Poisson structure ${\bf R}$
on ${\bf {R}}^3$ corresponds to a one-form $\alpha =A_{j}\,dx^{j}$
which can be written as $\alpha =u\, d\varphi$ for smooth
functions $u, \varphi$ defined on an open and dense subset of
${\bf R}^{3}$. The most general Poisson bracket in ${\bf {R}}^3$
can be therefore written in the form \be \{ x_i,x_j\} =\varepsilon
_{ijk}u({\bf x})\,{{\partial \varphi ({\bf x})} \over \partial
x_k}.\ee}
\newline
{\bf Remark}. It is worth noting that we really have to admit not
everywhere defined (singular) functions. For example, the bracket
in ${\bf {R}}^3$ defined by $\{ x_1,x_3\} =x_1,$ $\{ x_2,x_3\}
=x_2,$ $\{ x_1,x_2\} =0,$ corresponds to the one-form $\alpha
=x_2{\rm d}x_1-x_1{\rm d}x_2.$ Here the integrating factor is
$(x_1^2+x_2^2)^ {-1}$, so we can write \be \alpha =
(x_1^2+x_2^2)\,{\rm d}\left({\rm arctg}\,{x_1\over x_2}\right).\ee
We list now some relevant examples.
\bigskip
\noindent
\begin{enumerate}
\item[\bf I.] Assume that $u=1.$
     \begin{enumerate}
     \item[\bf 1.] The function  $\varphi $ is linear in $x_1, x_2, x_3$.
Then after orthogonal transformation we may consider $\varphi =
ax_3$ and we obtain \be \{x_1, x_2\} = a,\quad \{x_3, x_1\} =
\{x_3, x_2\} = 0.\ee We may take $x_3 = c$ and we have the
standard Poisson bracket.

      \item[\bf 2.] The function $\varphi $ is quadratic in $x_1, x_2, x_3$.
 Then by means of the orthogonal transformation we obtain
\be \varphi = {1\over 2}(a_1x_1^2 + a_2x_2^2 + a_3x_3^2).\ee We
have \be \{x_1, x_2\} = a_3x_3,\quad \{x_2, x_3\} = a_1x_1,\quad
\{x_3, x_1\} = a_2x_2.\ee

Let us consider some special cases:
            \begin{enumerate}
            \item[(i)]  \be a_1 = a_2 = a_3 = 1;\ee
we get the KKS-bracket corresponding to the Lie algebra $so(3).$

             \item[(ii)]  \be a_1 = a_2 = -a_3 = 1;\ee
we get the KKS-bracket corresponding to the Lie algebra $so(2,
1).$

             \item[(iii)]  \be a_{1}=a_{2}=1;\quad a_3 = 0;\ee
             \be \{x_1, x_2\} = 0,
\quad \{x_2, x_3\} = x_1,\quad \{x_3, x_1\} = x_2;\ee we get the
KKS-bracket corresponding to the Lie algebra $e(2).$

             \item[(iv)]  \be a_{1}=-a_{2}=1;\quad a_{3}=0,\ee
we get the KKS-bracket corresponding to the Lie algebra $e(1,1)$
\be \{ x_{1}, x_{2}\}=0,\quad \{ x_{2}, x_{3}\}=x_{1},\quad \{
x_{3}, x_{1}\} =-x_{2}.\ee

             \item[(v)]  \be a_1 = a_2 = 0,\ee
we get the KKS-bracket corresponding to the Heisenberg-Weyl
algebra ${\cal W}_1$ \be \{x_1, x_2\} = x_3,\quad \{x_2, x_3\} =
\{x_3, x_1\} = 0.\ee
           \end{enumerate}

        \item[\boldmath $2'.$]
When $\varphi=x_1x_2+x_2x_3+x_1x_3,$ the quadratic form associated
with $\varphi$ can be reduced to the canonical form
$\varphi=-2y_1^2+y_2^2+y_3^2$ and we get again $so(2,1).$

\item[{\boldmath $2''.$}] In the examples (i)-(v) we take $u(x)=1$ and obtain
the linear Poisson brackets. There are also four linear Poisson
brackets for the case $u(x)\neq {\rm const}$. This gives the
complete list of three-dimensional Lie algebras, obtained in the
previous century by L.~Bianchi.

         \item[\bf 3.] The function $\varphi $ is cubic.
Here we have 10 monomials, i.e.~in the general case, we have a
10-parameter family of brackets. We give only two examples.
         \begin{enumerate}
               \item[(i)] $\varphi = x_1x_2x_3.$ We get the bracket
\be \{x_1, x_2\} = x_1x_2,\quad \{x_2, x_3\} = x_2x_3,\quad \{x_3,
x_1\} = x_3x_1.\ee

               \item[(ii)] $\varphi ={1\over 3}(x_1^3+x_2^3+x_3^3).$
We get the bracket \be \{x_1, x_2\} = x_3^2,\quad \{x_2, x_3\} =
x_1^2,\quad \{x_3, x_1\} = x_2^2. \ee
                \end{enumerate}
        \end{enumerate}

By using a polynomial function of degree $k$ we get a polynomial
bracket of degree $k-1.$
\bigskip
\noindent

\item[\bf II.]
     \begin{enumerate}
     \item[\bf 1.] Put now $\alpha =u\,d\varphi ,$ where
     \be\begin{array}{rcl}
 \varphi &=&\psi (x_1)+\psi (x_2)+\psi (x_3),\\
 u&=& \chi (x_1)\chi (x_2)\chi (x_3),\\
\chi (x)&=&(\psi '(x))^{-1}.
\end{array}\ee
We get \be \{x_1, x_2\}=\chi (x_1)\chi (x_2),\,\,\{x_2, x_3\}=
\chi (x_2)\chi (x_3),\ee \nonumber \be \{x_3, x_1\}=\chi (x_3)\chi
(x_1).\ee
For example, if $\psi (x)=(1-m)^{-1}x^{1-m},$ we find
\be u\,d\varphi =(x_2x_3)^m{\rm d}x_1+(x_3x_1)^m{\rm
d}x_2+(x_1x_2)^m{\rm d}x_3.\ee

       \item[\bf 2.] Here \be \varphi =\psi (x_1)\,\psi (x_2)\,\psi (x_3)\,
\left({1\over \psi (x_1)}+{1\over \psi(x_2)}+{1\over \psi
(x_3)}\right)\ee and $u$ is as above.

Now \be u\,d\varphi =\left({\psi (x_2)\over \psi '(x_2)}\,{1\over
\psi '(x_3)}+ {\psi (x_3)\over \psi '(x_3)}\,{1\over \psi
'(x_2)}\right)\,{\rm d}x_1+ {\rm cyclic}.\ee

For instance, with $\psi (x)=(1-m)^{-1}x^{1-m}, $ we get \be
u\,d\varphi =c\,(x_2x_3^m+x_3x_2^m)\,{\rm d}x_1+{\rm cyclic}.\ee

\item[\bf 3.] For
\be u\,d\varphi =f_1(x_1,x_2)\,{\rm d}x_1+f_2\,(x_1,x_2)\,{\rm d}x_2\ee
the Poisson bracket is
\be \{x_1, x_2\} = 0,\quad \{x_2, x_3\} = f_1,\quad \{x_3, x_1\} = f_2.\ee

        \item[\bf 4.] $u\,d\varphi =F(x_1,x_2,x_3)\,{\rm d}x_3$ gives
\be \{x_1, x_2\} = F,\quad \{x_2, x_3\} = 0,\quad \{x_3, x_1\} =0.\ee
         \end{enumerate}
\end{enumerate}

\setcounter{equation}{0}
\subsection{Examples of three-dimensional Hamiltonian systems}

In this section, we give some examples of three-dimensional
Hamiltonian systems related to nonstandard Poisson structures. For
more details on these systems see Ref.\,\protect\citebk{[GN 1993]} and, especially, very
informative and clearly written book of Goriely.\cite{[Go 2001]}

\subsubsection{Euler top}

The Poisson structure for this system is related to the Lie
algebra $so(3)$ \be \{x_{1},x_2\}=x_3,\quad \{x_2,x_3\}=x_1,\quad
\{x_3,x_1\}=x_2, \ee and the Hamiltonian is quadratic in $x_j$:
\be H={1\over
2}\,(a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}).\ee Then
Hamilton's equations have the form: \be \dot
x_1=(a_{3}-a_{2})\,x_{2}x_{3},\quad \dot
x_2=(a_{1}-a_{3})x_{3}x_{1}, \quad \dot
x_3=(a_{2}-a_{1})\,x_{1}x_{2}\ee and coincide with Euler's famous
equations describing the rotation of a rigid body around a fixed
point.

\subsubsection{Lotka-Volterra system}

This system was considered first by Lotka\,\cite{[Lo 1920]} and 
Volterra.\cite{[Vo 1931]}
According to Ref.\,\protect\citebk{[GM-ORSW 1990]} we have
\begin{eqnarray}
\dot x & = & x\,(cy+z+\lambda),\\
\dot y & = & y\,(x+az+\mu),\\
\dot z & = & z\,(bx+y+\nu).\end{eqnarray}
\bigskip
Two cases are known with two functionally independent integrals of
motion:
\begin{description}
\item[i.]
\be \ abc = -1,\quad \nu =\mu b-\lambda ab\ee 
and
\item[ii.]
\be a=b=c=1,\quad \lambda =\mu =\nu =0.\ee
\end{description}

In the first case, the conserved quantities have the form:
\begin{eqnarray}I_1 & = & ab\,{\rm ln}\,x-b\,{\rm ln}\,y+{\rm ln}\,z,\\
I_2 & = & abx+y-az+\nu\,{\rm ln}\,y-\mu \,{\rm
ln}\,z\end{eqnarray} and this system is Hamiltonian with respect
to the Poisson structure \be \{x,y\}=cxy,\quad \{y,z\}=-yz,\quad
\{z,x\}=-bcxz.\ee

The second case is the limiting case of the Halphen system:
\begin{eqnarray}
\dot x & = & x(y+z),\nonumber \\
\dot y & = & y(z+x),\\
\dot z & = & z(x+y),\nonumber \end{eqnarray} and conserved
quantities have the form: \be I_1 = (xyz)^{1/2}\,\left({1\over
x}-{1\over y}\right),\quad I_2 = (xyz)^{1/2}\,\left({1\over
y}-{1\over z}\right).\ee This system is Hamiltonian with respect
to the Poisson structure \be \{x,y\} = xy\,(x-y),\quad
\{y,z\}=yz\,(y-z),\quad \{z,x\}=zx\,(z-x)\ee and the Hamiltonian
has the form \be H = {\rm ln}\,\left\{ \left[
\frac{z}{xy}\,(x-y)^2\right]^{\alpha }\, \left[
\frac{x}{yz}\,(y-z)^{2}\right]^{1-\alpha }\right\},\ee where
$\alpha $ is an arbitrary constant.

%$$A = (yz\,(y-z),\quad zx\,(z-x),\quad xy\,(x-y))$$

\subsubsection{Halphen system}

This system has the form:
\begin{eqnarray} \dot x & = & yz-x\,(y+z),\nonumber \\
\dot y & = & zx-y\,(z+x),\\
\dot z & = & xy-z\,(x+y).\nonumber\end{eqnarray} It is equivalent
to the system \be {{\rm d}\over {\rm d}t}\,(x+y)=xy,\quad {{\rm
d}\over {\rm d}t}\,(y+z)=yz, \quad {{\rm d}\over {\rm d}t}\,
(z+x)=zx,\ee which was considered first by Halphen.\cite{[Ha 1881]} This
system has been investigated extensively recently because of its
relation to the two-monopole system.\cite{[AH 1988]} After the change of
variables \be x=bc,\quad y=ac,\quad z=ab\ee we obtain the system
\begin{eqnarray}
\frac{2}{a}\,\frac{{\rm d}a}{{\rm d}t}=a^{2}-(b-c)^{2},\nonumber \\
\frac{2}{b}\,\frac{{\rm d}b}{{\rm d}t}=b^{2}-(c-a)^{2},\\
\frac{2}{c}\,\frac{{\rm d}c}{{\rm
d}t}=c^{2}-(a-b)^{2}.\nonumber\end{eqnarray} The solution of
equations (16) gives all $SO(3)$-invariant anti-self-dual Einstein
metrics. These equations  are derived from the metric of relative
mono\-pole space, i.e.~the space of motion of two monopoles with
respect to their center of mass, which is a four-dimensional
irreducible hyperk\"ahler manifold.\cite{[AH 1988]} In this case it may
be shown\,\cite{[GN 1993]} that the system cannot be globally completely
integrable.

%$$A_{x} = (x\,(y^{2}-z^{2})+yz\,(z-y))$$

\subsubsection{Lorenz model}

This model was introduced in Ref.\,\protect\citebk{[Lo 1963]}
and it is described by the equations
\begin{eqnarray}
\dot x & = & \sigma \,(y-x),\nonumber \\
\dot y & = & (\rho x-xz-y),\\
\dot z & = & (-\beta z+xy),\nonumber \end{eqnarray} where $\sigma
$ and $\rho $ are the Prandtl and Rayleigh numbers, respectively.

This system has a chaotic behavior for most values of parameters
$\sigma , \rho $ and $\beta $, however, it becomes Hamiltonian for
two limiting cases. The first of them is \be \rho = 0,\quad \sigma
={1\over 2},\quad \beta =1.\ee In this case, the quantity \be
I={{y^{2}+z^{2}}\over {(x^{2}-z)^{2}}}\ee is the integral of
motion and this system is the Hamiltonian with respect to the
Poisson structure \be\{y,z\} = -2(y^{2}+z^{2})\,x,\quad
\{z,x\}=(x^{2}-z)\,y,\quad \{x,y\}=x^{2}z+y^{2}.\ee

\subsubsection{Kermack-McKendrick models for epidemics}

These models describe the spread of epidemics for a closed
population and have the form\,\cite{[KK 1927],[KK 1932],[KK 1933]}
\be \dot S=-rSI,\quad \dot I=rSI-aI,\quad \dot R=aI,\ee
where $S$ denotes the number of susceptibles, $I$ the number of
infected and $R$ the number of removals. The constants $a,r$
determine the infection and removal rates of infectives,
respectively.

The equation of motion here admits two integrals of motion
\begin{eqnarray}
I_1 & = & S+I+R,\nonumber \\
I_2 & = & R+\frac{a}{r}\,{\rm log}\,S\nonumber \end{eqnarray}
which form the Hamiltonian system relative to the Poisson
structure
\be \{S,I\} = SI,\quad \{I,R\} = SI,\quad \{R,S\} = SI.\ee

\subsection{The four-dimensional case}
\setcounter{equation}{0}

Let us consider the simplest case of four-dimensional Euclidean
space ${\bf R}^{4}$. In this case, the Poisson structure is
determined by the antisymmetric tensor  $\omega ^{\mu \nu }$ or by
the vector $\Lambda $: \be \omega ^{\mu \nu }=\{x^{\mu },x^{\nu
}\},\,\,\Lambda =\omega ^{\mu \nu }
\partial _{\mu }\land \partial _{\nu },\,\,\partial _{\mu }={\partial \over
{\partial x^{\mu }}},\,\,\mu ,\nu =1,\ldots ,4.\ee Following 
Ref.\,\protect\citebk{[GMP 1993]} we should consider two different cases.
\begin{description}
\item[I.] ~~~Tensor $\omega ^{\mu \nu }$ is non-degenerate
(i.e.~${\rm det}\,(\omega ^{\mu \nu })\neq 0$) on some open dense
submanifold of ${\bf R}^{4}$.

On this submanifold we may consider the inverse tensor $F_{\mu \nu}$
and the two-form
\be F = F_{\mu \nu }\,dx^{\mu }\land dx^{\nu}.\ee
We may also write this form analogously as in
electrodynamics
\be F =  (E_{j}\,dx^{j})\land
dx^{4}-B_{jk}\,dx^{j}\land dx^{k},\qquad j,k=1, \ldots ,3.\ee
If we introduce $H^{k}=\varepsilon ^{ijk}B_{jk}$ we may also rewrite
$F$ in terms of the vector ${\bf H}$. We have \be F\land F=2{\bf
EH}\,dx^{1}\land dx^{2}\land dx^{3}\land dx^4.\ee Therefore our
bivector field will be degenerate at the points where ${\bf
EH}=0$. It is easy to show that in the case under consideration,
the Jacobi identity is equivalent to the equation
\be dF = 0,\ee
or, in other words, the vectors ${\bf E}$ and ${\bf H}$ should
satisfy the Maxwell equations.

Thus we may express the tensor $F_{\mu \nu }$ in terms of the
vector potential $A_{\mu }$: \be F_{\mu \nu }=\partial _{\mu
}A_{\nu }-\partial _{\nu }A_{\mu },\ee which is determined up to
the ``gauge'' transformation \be A_{\mu }\to A_{\mu }+\partial
_{\mu }f.\ee Therefore in this case, the Poisson structure is
determined by three real functions of four variables.

\item[II.] ~~The tensor $\omega ^{\mu \nu }$ is completely degenerate:
\be {\rm det}\,(\omega ^{\mu \nu })\equiv 0.\ee The general
Poisson structure for this case is given by the theorem of the
paper.\cite{[GMP 1993]}
\newline  \newline
{\bf Theorem 2.} {\em If the Poisson structure on ${\bf R}^{4}$ is
degenerate, i.e.~${\rm det}\,(\omega ^{\mu \nu })\equiv 0$, it is
given by the formula \be \omega ^{\mu \nu }=\{x^{\mu }, x^{\nu
}\}= u(x)\,\varepsilon ^{\mu \nu \lambda \sigma }\partial
_{\lambda }C_{1}(x)\,\partial _{\sigma } C_{2}(x).\ee where the
functions $u(x),\,C_{1}(x)$ and $C_{2}(x)$ are defined on an open
and dense submanifold in ${\bf R}^{4}$.}

Note that in general, the exact forms ${\rm d}C_1$ and ${\rm
d}C_2$ can be replaced by closed one-forms.
\end{description}
\bigskip
We close these general considerations by saying that whenever $F$
reduces to a pure ``electric", or ``magnetic", or ``radiation"
field, we will have the degenerate Poisson bracket.

\noindent Let us give some examples.\\

\begin{itemize}
\item{\bf Example 6}.
Due to Theorem 2, the form \be F={\rm d}(a_{\mu }x_{\mu }^2)\wedge
{\rm d}(b_{\nu }x_{\nu }^2)\ee corresponds to the Poisson
structure $\Lambda$ on ${\bf {R}}^4.$ Since \be F=2(a_{\mu }b_{\nu
}-b_{\mu }a_{\nu })\,x_{\mu }x_{\nu }\,{\rm d}x_{\mu } \wedge {\rm
d}x_{\nu },\ee so \be \Lambda =\varepsilon _{\mu \nu \lambda
\sigma }\,(a_{\mu }b_{\nu }- b_{\mu }a_{\nu })\,x_{\mu }x_{\nu
}\,\partial _{\lambda }\wedge \partial _ {\sigma },\ee and we get
the the covariant form of Poisson bracket \be \{ x_{\mu },x_{\nu
}\} =\varepsilon _{\mu \nu \lambda \sigma }(a_ {\lambda }b_{\sigma
}-b_{\lambda }a_{\sigma })\,x_{\lambda }x_{\sigma }.\ee In
particular case $b_4=0,\ b_1=b_2=b_3=1, \ a_4=1,$ we get the
bracket considered by Sklyanin\,\cite{[Sk 1982]} \be \{ x_k,x_l\}
=\varepsilon _{jkl}\,x_4x_j,\quad \{ x_j,x_4\} =\varepsilon
_{jkl}\,(a_k-a_l)\,x_kx_l,\ee where $j,k,l=1,2,3.$

\item{\bf Example 7}.
Put \be F=(E_idx^i)\wedge dx^0-\varepsilon _{ijk}\,H_i\,dx^4\wedge
dx^k.\ee Then \be \Lambda = \partial _4\wedge (H_i\partial
_i)+\varepsilon _{ijk}\, E_k\,\partial _i\wedge \partial _j,\ee
i.e. \be \{ x^4,x^i\} =H_i, \ \{ x^i,x^j\} =\varepsilon
_{ijk}E_k.\ee Let us look for quadratic brackets. From $F={\rm
d}A$ and the quadratic requirement, $A=A_i{\rm d}x^i$ must have
$A_i$ of degree three. Therefore \be {\rm d}(x_4(x_i^2{\rm
d}x_i)+(x_1^3+x_2^3+x_3^3){\rm d}(x_1+x_2+x_3))=F\ee will provide
us with an instance of a quadratic bracket in four dimensions. It
is possible to give many more examples. For instance, by taking
$C_1=x_1x_4-x_2x_3,\ \ C_2=x_1^2+...+x_4^2$ we again get the
quadratic Poisson bracket. It should be noticed that if we
consider the Ampere two-form
$$G=(H_i{\rm d}x^i)\wedge {\rm d}x^4+D_{ij}{\rm d}x^i\wedge {\rm d}x^j$$
with $G\wedge G\ne 0$, the associated bivector field will satisfy
the Jacobi identity except at the world lines of the sources. For
the electromagnetic field of a moving charge the Jacobi identity
fails only along the world line of the charge. This situation
generalizes into four dimensions the situation for magnetic
monopoles in ${\bf {R}}^3.$

\item{\bf Example 8}.
Let us take
$$u(x)=x_{3}^{2}+x_{4}^{2},\,\,C_{1}={1\over 2}\,(x_{1}^{2}+x_{2}^{2}+
x_{3}^{2}+x_{4}^{2}),\,\,C_{2}={\rm arctg}\,{x_{3}\over x_{4}}.$$
The corresponding Poisson bracket in ${\bf {R}}^4$ is quadratic
and has the form \be \{ x_2,x_1\} =x_3^2+x_4^2;\qquad \{ x_1,x_4\}
=x_2x_4;\qquad \{ x_1,x_3\} =x_2x_3;\ee \be \{ x_2,x_4\}
=-x_1x_4;\qquad \{ x_2,x_3\} =-x_1x_3;\qquad \{ x_3,x_4\} =0.\ee
This bracket is invariant with respect to the left and right
action of the group of matrices
\begin{displaymath}
\left(\begin{array}{cc} e^{{\rm i}t} & 0\\ 0 & e^{-{\rm i}t}
\end{array} \right)
\end{displaymath}
and is related to the so-called Lie-Poisson structure on the group
$G={\bf R}_{+}\otimes SU(2)$ (see Ref.\,\protect\citebk{[GMP 1993]}) 
-- group of $2 \times 2$
matrices of the type
\begin{displaymath}
\left(\begin{array}{cc} x_1 + {\rm i}x_2 &-x_3 +{\rm i}x_4 \\
x_3 +{\rm i}x_4 & x_1 -{\rm i}x_2 \end{array} \right)
\end{displaymath}

\item{\bf Example 9}.
As above, if we take $u(x)=(x_{2}^{2}+x_{3}^{2}),\,\,C_{1}=x_{1}
x_{4}-x_{2}x_{3}$ and $C_{2}={\rm arctg}\,(x_{2}/x_{3})$, then we
get the Poisson brackets:
 \be \{ x_2,x_1\} =x_2x_1;\quad
\{ x_1,x_4\} =-2x_2x_3;\quad \{ x_1,x_3\} =-x_1x_3;\ee \be \{
x_2,x_4\} =-x_2x_4;\quad \{ x_4,x_3\} =x_4x_3;\quad \{ x_3,x_2\}
=0.\ee These brackets are invariant with respect to the left and
right action of the group of matrices
\begin{displaymath}
\left(\begin{array}{cc} e^{t} & 0\\ 0 & e^{-t} \end{array} \right)
\end{displaymath}
and are related to the Lie-Poisson structure on the group
$G=GL\,(2,{\bf R})$.
\end{itemize}

At the end of this section we mention the paper\,\cite{[CIMP 1994]}
where the Poisson structure approach was used for classification
of low-dimensional Lie algebras.

\subsection{Generalized Poisson structures} In this section we just
mention without going into any detail two generalizations of
Poisson structures.
\bigskip

\noindent {\bf 1.}\, At 1973 Nambu\,\cite{[Na 1973]} proposed the
generalization of the standard classical Hamiltonian mechanics
based on a three�dimensional `phase space' spanned by a canonical
triplet of dynamical variables and on two `Hamiltonians' $h_1$ and
$h_2$. For $n$-dimensional space the dynamics of the system is
determined by $n-1$ Hamiltonians. The typical equations of motion
are of the form
$$
\dot f =\frac{D(f,h_1,...,h_{n-1})}{D(x_1,x_2,...,x_n)}
$$
where the r.h.s. is the Jacobian. For more details on Nambu
mechanics see Refs.\,\protect\citebk{[Ta 1994]} 
and \protect\citebk{[AG 1996]}. In the last
paper the important theorem was proven that the general Nambu case
may be reduced to several typical cases.

\bigskip

\noindent {\bf 2.}\, Another generalization of standard Poisson
brackets was given in papers by  Azc\'arraga, Perelomov and
P\'erez Bueno.\cite{[APP 1996a],[APP 1996b]} Let us consider here the
special case of this bracket when the space ${\Bbb R}^n$ is
endowed by the structure $n$-dimensional compact simple Lie
algebra $\cal G$ with the orthonormal basis $e_i$ relative the
Killing-Cartan form. So we have
$$
[e_i,e_j]=c_{ijk} e_k
$$
where the tensor $c_{ijk}$ is totally antisymmetric one.

The generalized Hamilton's equations here have the form.
$$
\dot x_{i}= c_{ij_{1}...j_{k-1}} \partial _{j_{1}}h_1 ...\partial
_{j_{k-1}}h_{k-1}
$$
where $c_{ij_{1}...j_{k-1}}$ is $\cal G$-invariant totally
antisymmetric tensor (which may be considered as an element of
$k$-th cohomology space ${\cal H}^k$ ).

Note that these brackets are generalized to the case of Lie
superalgebras (see Ref.\,\protect\citebk{[AIPP 1977]} 
where interesting reader 
may find the details).

%%%%%%%%%%%%

%\newpage

\setcounter{equation}{0}
\section{Explicit Integration of Equations of Motion}
\subsection{Projection method}

Let us recall first of all that the Hamiltonian system with $n$
degrees of freedom will be completely integrable if there are $n$
integrals of motion in involution \be \left\{ I_j, I_k\right\} =
0\ \ ,\quad j,k = 1,\ldots, n\ \ ,\quad I_1 =
 H,\ee
which are functionally independent.

In this case, the Bour-Liouville-Arnold 
theorem\,\cite{[Bou 1855],[Lio 1855],[Ar 1989]} states that the motion 
may be reduced to the
motion on a torus (or on Euclidean space). Unfortunately, this
theorem does not give the recipe for explicit integration of the
equation of motion.

In the standard approach--the so-called Hamiltonian reduction 
(Jacobi (around 1830), Lie\,\cite{[Lie 1874/75]} and 
Marsden--Weinstein\,\cite{[MW 1974]}) one try to use the integrals 
of motion to decrease the
number of degrees of freedom.  But the dynamical behavior on the
reduced manifold becomes more complicated and except of some
simple cases this method does not help to integrate the equations
of motion.

The main idea of the projection 
method\,\cite{[OP 1976b],[OP 1976c],[Pe 1992]} is to inverse the process 
of the Hamiltonian reduction.
Suppose we have a dynamical system on the manifold $M=\{x\}$ which
generate the flow
$$
g_t: M \to M,\quad x_t=g_t x_0 .
$$
Let us try to find another system with a larger number of degrees
of freedom $N>n$ on the another manifold $\tilde M=\{y\}$
$$ \tilde g_t: \tilde M \to \tilde M,\quad y_t=\tilde g_t y_0$$
so that the following conditions are satisfied:
\begin{enumerate}
\item[{\bf (i)}] The dynamical behavior of the new system
is sufficiently simple
so that the equation of motion of this system may be integrated explicitly.

\item[{\bf (ii)}] The initial system is the projection $\pi$ of the larger one
$$
\pi : \tilde M \to M
$$
and this projection may be described by explicit formulae.

%\newpage
\item[{\bf (iii)}] There exists a map $\rho : M \to \tilde M$ such that
$\pi \cdot \rho = {\rm id} $ which may be described explicitly.
\end{enumerate}

In this case, we may integrate the equations of motion for the
Hamiltonian system on $M$ explicitly.  Indeed, we have \be x_t =
\pi \cdot y_t = \pi \cdot \tilde g_t \cdot y_0 = \pi \cdot \tilde
g_t \cdot \rho\cdot x_0, \ee
i.e. \\
\be \gboxit{x_t = \pi \cdot \tilde g_t \cdot \rho \cdot x_0}\ \ .\ee

The main difficulty with this method is in finding $\tilde M$,
$\pi$, $\tilde g_t$ and $\rho$.  We do not know how to do this in
general, but we have a number of interesting and nontrivial
examples of application of this method. We consider some of them
below.

\subsection{Systems of Calogero--Sutherland type}

\setcounter{equation}{0}

These are one-dimensional systems (systems of $n$ particles on a
line) interacting pairwise via potentials $V(q) = g^{2}\,v(q)$ of
the following 5 types\,\footnote{\,For more details on these
systems see Ref.\,\protect\citebkcap{[OP 1981a]} and 
books.\cite{[Pe 1990],[Ca 2001]}}
 \be \begin{array}{rl}
{\rm I.}\quad \quad \quad & v(q) = q^{-2},\\ \\
{\rm II.}\quad \quad \quad & v(q) = a^{2}\,\sinh ^{-2}\,aq,\\ \\
{\rm III.}\quad \quad \quad & v(q) = a^{2}\,\sin ^{-2}\,aq,\\ \\
{\rm IV.}\quad \quad \quad & v(q) = a^{2}\,{\cal P}(aq),\\ \\
{\rm V.}\quad \quad \quad & v(q) = q^{-2}+\omega ^{2}q^{2}.
\end{array} \ee

Here ${\cal P}(q)$ is the Weierstrass function, i.e.~the
meromorphic double-periodic function with periods $2\omega _{1}$
and $2\omega _{2}$ in a complex plane and asymptotic behavior near
the pole at point $q_0$ \be {\cal P}(q) \sim
(q-q_{0})^{-2}+c_{2}\,(q-q_{0})^{2}+c_{3}\,(q-q_{0})^{4} +\ldots
.\ee The investigation of systems of type~I was initiated by
F.~Calogero\,\cite{[Ca 1971]} and of type~III by 
B.~Sutherland\,\cite{[Su 1971]}
(it was continued in Ref.\,\protect\citebk{[Ca 1975]}). 
It was later proved by 
J.~Moser\,\cite{[Mo 1975]} that the systems of type I are completely 
integrable in
the Bour-Liouville-Arnold sense.

To prove complete integrability it is convenient to use the Lax
representation. We shall first exhibit the Lax pairs for the
Calogero-Sutherland systems of types I-IV, which will allow to
analyze the integrability of these systems. Next we shall consider
systems of type V.

The Lax pairs for systems of types I-IV were found by 
Moser\,\cite{[Mo 1975]} and by Calogero.\cite{[Ca 1975]} 
They are given by the matrices:
\be L_{jk} = p_{j}\delta _{jk}+{\rm i}g\,(1-\delta
_{jk})\,x\,(q_{j}-q_{k}),\ee \be M_{jk} = g\,\Bigl [\delta
_{jk}\,\sum _{l\neq j}\,z\,(q_{j}-q_{l})-(1-\delta
_{jk})\,y\,(q_{j}-q_{k})\Bigr ].\ee
Substituting these matrices into the Lax equation
$$\dot L = [L,M]$$
and requiring this equation to be equivalent to the original
Hamilton equations we find both \be y(\xi) = x'(\xi)\ee and a
functional relation for $x(\xi)$ and $z(\xi)$: \be
x'(\xi)\,x(\eta)-x'(\eta)\,x(\xi) = x(\xi
+\eta)\,[z(\xi)-z(\eta)]. \ee The function $v(\xi)$ which occurs
in the potential energy is given by the formula: \be v(\xi) =
x(\xi)\,x(-\xi)+{\rm const}.\ee The functional equation (6) was
solved in several papers.\cite{[Ca 1976],[OP 1976a],[PS 1976]} It
turns out that \be z(\xi) = x''(\xi)/2x(\xi),\ee and all odd
solutions $x(\xi)$ have the form: \be \begin{array}{l} x(\xi) =
\left\{
\begin{array}{lll}
\xi ^{-1}&&{\bf I}\\
a\,\coth \,a\xi ,&a\,\sinh ^{-1}\,a\xi &{\bf II}\\
a\,\cot \,a\xi ,&a\,\sin ^{-1}\,a\xi ,&{\bf III}\\
a\,{{{\rm cn}\,a\xi }/ {{\rm sn}\,a\xi }},&a\,{{{\rm dn}\,a\xi }/
{{\rm sn}\,a\xi }},\,\,{a / {{\rm sn}\,a\xi }}&{\bf IV}
\end{array} \right. \end{array} \ee Here ${\rm cn}\,(\xi),\,\,{\rm
sn}\,(\xi)$ and ${\rm dn}\,(\xi)$ are the Jacobi elliptic
functions.

It is easy to verify, by using (7), that the potentials of types
I-IV in~(1) result from these expressions. Substituting ${\rm i}a$
for $a$ we take the function $x(\xi)$ of type II to the function
$x(\xi)$ of type III. In the limit as $a\to 0$, the function
$x(\xi)$ of type II or III goes over into $x(\xi)$ of type I. The
same relationship holds for the matrices $L$.

The Lax representations for more general systems of types I-IV,
the so-called systems related to other classical root systems (for
details, see Ref.\,\protect\citebk{[Pe 1990]}) were found by Olshanetsky 
and Perelomov.\cite{[OP 1976a]}

In the general case, both matrices $L$ and $M$ have dimension
$2n+1$ and can be written in the form: \be L = P+X,\quad
M=-D+Y,\ee where
\begin{eqnarray}
P &=& {\rm diag}\,(p_{1},\ldots ,p_{n},\,0\,-p_{n},\ldots ,\,-
p_{1});\nonumber\\
X &=& \left| \begin{array}{ccc}
A_{1}&C_{1}^{T}&B_{1}\\
\overline {C}_{1}&0&-\overline {C}_{1}\\
-B_{1}&-C_{1}^{T}&-A_{1}\end{array} \right|, \nonumber\\
&&\\
Y &=& \left| \begin{array}{ccc} A_{2}&C_{2}^{T}&B_{2}\\
-\overline {C}_{2}&0&-\overline {C}_{2}\\
B_{2}&C_{2}^{T}&A_{2}\end{array}
\right|, \nonumber\\
D &=& {\rm diag}\,(d_{1},d_{2},\ldots
,d_{n},\,d_{0},\,d_{n},\ldots ,d_{2}, d_{1}),\nonumber
\end{eqnarray}
and where $A_{1}, B_{1}, A_{2}, B_{2}$ are $n\times n$ matrices:
\begin{eqnarray}
\{ A_{1}\}_{kl} &= & {\rm i}(1-\delta _{kl})\,gx(q_{k}-q_{l},\nonumber\\
\{ B_{1}\}_{kl} &= & {\rm i}g_{2}\,\sqrt {2}\,\delta
_{kl}\,x(2q_{k})+{\rm i}g\,(1-\delta
_{kl})\,x(q_{k}+q_{l}),\nonumber\\
\{ A_{2}\}_{kl} &= & {\rm i}(1-\delta _{kl})\,gy\,(q_{k}-q_{l}),\nonumber\\
\{ B_{2}\}_{kl} &= & {\rm i}g_{2}\,\sqrt {2}\,\delta
_{kl}\,y(2q_{k})+{\rm i}g\,(1-\delta
_{kl})\,y(q_{k}+q_{l}),\nonumber\\
\{ C_{1}\}_{k} &= & {\rm i}g_{1}\,x(q_{k}),\\
\{ C_{2}\}_{k} &= & {\rm i}g_{1}\,y(q_{k})\nonumber\\
d_{k} &= & {\rm i}(e_{k}-c),\nonumber\\
e_{k} &= & \sum _{r=1}^{n}\,g[z(q_{k}-q_{r})+z(q_{k}+q_{r})] +
g_{1}^{2}g^{-1}
\,z(q_{k})+\sqrt {2}\,g_{2}z(2q_{k}),\nonumber\\
e_{0} &= & 2g\,\sum _{k=1}^{n}\,z(q_{k}),\quad c = {2\over
{2n+1}}\,\Bigl [ \sum _{k=1}^{n}\,e_{k}+{1\over 2}\,e_{0}\Bigr
].\nonumber
\end{eqnarray}

These Lax pairs will be valid only if the interaction constants
obey the following relations:
$$ g_{1}^{2}+\sqrt {2}g_{1}g_{2}-2g_{2}^{2}=0\quad {\rm if}\quad g_{1}=0,$$
$$g_{1}, g_{2}\quad \mbox{ arbitrary if }\quad g_{1}=0.$$

As before, the functions $x(\xi), y(\xi)$ and $z(\xi)$ satisfy (5)
and (6) so that the interaction potential has the form (1).

To show that these systems are completely integrable we consider
the invariants of $L$ \be I_{k} = {1\over k}\,{\rm tr}\,L_{k},\ee
which are obviously the integrals of motion. The matrix structure
of $L$ implies that these integrals have the form: \be I_{k} =
{1\over k}\,\sum \,p_{j}^{k} + \mbox{ terms of lower degree in the
momenta}.\ee Hence the functional independence of the $I_{k}$
follows from the functional independence of the quantities \be
s_{k} = \sum _{j} \,p_{j}^{k}.\ee Let us note that the polynomials
$s_{k}$ are invariants of the corresponding simple (or affine) Lie
algebra. The degrees of the basic invariants generating  the whole
algebra of invariants are well known. For the classical root
systems, they are:
\begin{eqnarray*}
A_{n-1}&:&k = 2,3,\ldots ,n,\\
B_{n}, C_{n}, BC_{n}&:&k = 2,4,\ldots ,2n,\\
D_{n} &:&k = 2,4,\ldots ,2(n-1),n.
\end{eqnarray*}

The involutivity of the integrals for systems of types I-III and
for the non-periodic Toda lattices can be proved by using the
argument of Calogero.\cite{[Ca 1975]} For this purpose, we consider
first the system of type II. Since the potential $U(q)$ is
repulsive we can show that $q_{a}(t)\to \infty $ as $t\to \infty
$. (For systems of type $A_{n-1}$, this means that the particles
will move infinitely far away from each other.)  In this case, as
is clear from the structure of $L$, the integral $I_{k}$ (14)
tends to $ks_{k}$ (15), and hence $\{I_{k},I_{j}\}\to
kj\{s_{k},s_{j}\}=0$. Since the Poisson bracket of any two
integrals is also an integral, this implies $\{I_{k},I_{j}\} = 0$.
Thus both the systems of type II $A_{n-1}$ and II $BC _{n}$ are
completely integrable.

The systems of types I and III are obtained from the systems of
type II by taking the limit as $a\to 0$ or by substituting
$a\mapsto {\rm i}a$, respectively, which implies their
integrability.

For the non-periodic Toda lattices, one can show that $q_{n}(t)\to
-\infty $ as $t\to \infty $. In particular, for the ordinary
lattice this means that $q_{j+1}(t)-q_{j}(t)\to \infty $. As
before, it follows that $I_{k}(t) \to s_{k}$, and the previous
reasoning remains valid.

It is more difficult to prove the involutivity of the integrals of
motion for systems of type IV. In this case, it is convenient to
take for the basic integrals the eigenvalues of $L$.\,\cite{[Pe 1977]}

For the systems of types I-IV(1) and the ordinary Toda lattice, it
is convenient, in addition to the integrals $I_{k}$ (13), to
consider another set of integrals $J_{k}$ which are the
coefficients of the characteristic polynomial of $L$: \be {\rm
det}\,(L-\lambda E) = \lambda ^{n}+\sum _{k=1}^{n}\,J_{k}\,
\lambda ^{n-k}.\ee The two integrals $I_{k}$ and $J_{k}$ are
related by Newton's formula: \be I_{k} = I_{k-1}J_{1} -
I_{k-2}J_{2}+\cdots -(-1)^{k-1}\,I_{1}J_{k-1} -
(-1)^{k}\,kJ_{k}.\ee

\subsection{Systems with external harmonic potential}
\setcounter{equation}{0}

A modification of the Lax equation\,\cite{[Pe 1976]} (see Appendix) allows
us also to treat systems with external harmonic potential (systems
of type V). In this case, the equations of motion are equivalent
to the matrix equations \be \dot L^{\pm } = [L^{\pm },M] \pm {\rm
i}\omega L^{\pm }, \ee where $M$ is the same as in the Lax pair
for the system of type I, and \be L^{\pm } = L\pm {\rm i}\omega
Q\ee with \be Q = {\rm diag}\,[q_{1},\ldots ,q_{n}] \ee for the
systems of type V.
To show the equivalence one uses the relation \be [Q,M] = X, \ee
where $X$ is the off-diagonal part of $L$.

From (1) it follows that the quantities
\be B_{k}^{\pm } = {1\over
k}\,{\rm tr}\,(L^{\pm })^{k} \ee
have a simple time dependence:
\be B_{k}^{\pm }(t) = B_{k}^{\pm }(0)\,\exp \,(\mp {\rm i}k\omega
t).\ee Using (1) we can also find the integrals of motion. For
this purpose, let us consider the matrices \be N_{1} =
L^{+}\,L^{-},\quad N_{2} = L^{-}\,L^{+}. \ee These satisfy the
usual Lax equation \be \dot N_{j} = [N_{j},M],\quad j=1,2, \ee
therefore the traces of powers of these matrices are the integrals
of motion.

These are a system of $n$ particles in line, with the Hamiltonian
\be H = {1\over 2} \sum\limits^k_1 p^2_j + g^2 \sum\limits_{j<k}
{1\over \left (q_j - q_k\right)^2}.\ee The phase space in this
case is standard: \be M = \left\{ \left({\bf p},{\bf q}\right)
\big| {\bf p} = \left( p_1,\ldots, p_n\right)\ \ ,\qquad{\bf q} =
\left(q_1, \ldots , q_n\right)\right\}.\ee Such a system was
considered\,\cite{[Ja 1866]} for $n=3$ by Jacobi\,\footnote
{\,Note that if a system with
$n$ degrees of freedom has $n$ functionally independent integrals
of motion in involution which depend on momenta quadratically,
then it may be solved by the St\"ackel separation  variable method
(see Refs.\,\protect\citebkcap{[Wh 1937]} and 
\protect\citebkcap{[Pe 1990]}).}, who showed 
it to be integrable. For
arbitrary $n$, such systems were investigated by Calogero around
1970.\cite{[Ca 1971],[Ca 1975],[CMR 1975]} The systems for $n=4$ and
$n=5$ were shown to be integrable,\cite{[Pe 1971]} but I did not succeed
at that time to prove that the system is integrable for arbitrary
$n$.  This was achieved in 1975 by Ju.~Moser\,\cite{[Mo 1975]}
 (see also Ref.\,\protect\citebk{[Mo 1980]}) who found the $L-M$ pair for 
this system:
   \be \dot L = [L,M].\ee
    The quantities
\be I_k = {1\over k} \tr \left(L^k_1\right),\qquad I_1 = \Sigma
p_j, \qquad I_2 = H,\ldots \ee are $n$ functionally independent
integrals of motion in involution \be \left\{ I_j, I_k\right\} = 0
\ee
so the Calogero system is a completely integrable Hamiltonian
system.

However, not all integrals of motion are quadratic in momenta and,
 hence, it is not clear how to integrate the equations of motion.

The equations of motion for the systems of types I, II, III, V and
for the generalized non-periodic Toda lattices can be solved
explicitly by means of the projection method described above. In
this section these examples will be extended to include the
systems with more than one degree of freedom.

\subsection{Integration of equations of motions for rational case}
\setcounter{equation}{0}

We consider here the systems of types I and V of 
Sec.\,{\em 2.2}.\,\cite{[OP 1976a]} The construction makes use of 
the space $X_{n}^{0} =
\{ x\}$ of $n\times n$ traceless complex Hermitian matrices which
can be viewed as the Euclidean space with a metric given by \be
{\rm d}s^{2} = {\rm tr}\,({\rm d}x)^{2}. \ee The equations of
geodesics in $X_{n}^{0}$  can be written in the Hamiltonian form.
This means that the phase space is $T^{*}X_{n}^{0}$, the
symplectic structure is determined by the 2-form $\omega $
($\omega ={\rm tr}\,({\rm d}x\land {\rm d}y$)), and the
Hamiltonian is $H={1\over 2}\,{\rm tr}\,(y^{2})$ where $y$ is the
momentum variable. The equation of the geodesic is
\be \ddot x=0\ee
and its solutions are the straight lines
\be x(t) = at+b.\ee
Clearly, both the metric (1) and the form $\omega ={\rm tr}\,
({\rm d}x\wedge {\rm d}y) $ are invariant under unitary
transformations $  x \to uxu^{+}$. Let us reduce the Hermitian
matrix $x(t)$ to the diagonal form $Q(t)$ \be x(t) =
u(t)\,Q(t)\,u^{-1}(t), \ee where \be Q(t) = {\rm
diag}\,[q_{1},\ldots ,q_{n}]. \ee We can assume that
$q_{1}>q_{2}>\ldots >q_{n}$. The map of $x$ to $Q$ is called the
{\em spherical projection}.

We shall derive the equations of motion for $Q(t)$ and $P(t)=\dot Q(t)$ on
the condition that $x(t)$ satisfies (2).
By differentiating (4) with respect to time we obtain
\be u(t)\,L(t)\,u^{-1}(t) = a \ee
with
\be L(t) = P(t)+[Q,M], \ee
\be M = -u^{-1}(t)\,\dot u(t), \ee
where $L$ is the Hamiltonian and $M$ is the skew-Hermitian matrix.
Differentiating (6) again we obtain
\be \dot L = [L,M]. \ee
Thus the pair of matrices  (7) and (8)
satisfies the Lax equation.

On the other hand, one can easily verify that the Lax matrices for the
systems of type I satisfy (7). This establishes the equivalence of
these two systems.

However, we must note that not every geodesic is mapped by the spherical
projection into a trajectory of the system of 
type I.\,\footnote{\,From mathematical point of view this means
 that we should fix the value of moment map, see for details 
Ref.\,\protect\citebkcap{[KKS 1978]}.}
Indeed, let
us consider an ``angular momentum" $\mu$ of the geodesic (3):
\be \mu = [x,\dot x] = [b,a] = u[Q,L]\,u^{-1}.\ee
It is easy to see that a commutator $[Q,L]$ has the rather special
structure:
\be [Q,L] = g\,(v^{T}\otimes v-I),\quad v= (1,1,\ldots ,1), \ee
i.e.~it has $n-1$ coinciding eigenvalues. (The symmetry group $G$ which
determines the momentum map $\mu $ is $SU(n)$; the subgroup $G_{\mu }$ which
leaves invariant the momentum $\mu $ (10) consists of the matrices in $SU(n)$
commuting with $[Q,L]$.)

Now we can obtain the solution of the Cauchy problem for systems of type~I.
We set $u(0)=I$ and reconstruct parameters of the geodesic from
the initial  data:
\be a = L(0),\quad b=Q(0). \ee
Using (3) we arrive at the important conclusion:\\
\noindent
{\em the coordinates $q_{j}(t)$ are the
eigenvalues of the matrix}
\be Q(0) + L(0)t. \ee

To determine the solutions of the equations of type V  we take the
same space $X_{n}^{0}$, but instead of free motion we consider the
equation of a harmonic oscillator:
\be \ddot x+\omega ^{2}x = 0. \ee
Its solutions are
\be x(t) = a\omega ^{-1}\,\sin \,\omega t + b\cos \,\omega t,\quad a,b\in
X_{n}^{0}.\ee
As before, taking the spherical decomposition (4) of $x(t)$ and
differentiating twice with respect to $t$ we are able to write the dynamical
system (9) in the form (14). The equivalence implies the important result:\\
\noindent
{\em the coordinates $q_{j}(t)$ of the trajectories
of the system of type V are the eigenvalues of the matrix}
\be Q(0)\,\cos \,\omega t + \omega ^{-1}\,L(0)\,\sin \,\omega t. \ee

These results can be used to derive many useful corollaries. Here
we give just two of them.

First, from them it immediately follows that rational systems are
superintegrable, i.e. these systems with $n$ degrees of freedom
have not only $n$ integrals of motion, but $2n-1$ integrals of
motion.

Second, the quantities $I_{jk}$ defined by the formula
$$
I_{jk} =\frac{1}{(j+k)!}\left({\frac{\partial}{\partial{\lambda}
}}\right)^k {\,\rm tr \,}(L + \lambda Q)^{j+k}\left|_{\lambda=0}
\right.$$
form very interesting infinite-dimensional Lie algebra relative
the Poisson brackets. The investigation of this algebra start at
Ref.\,\protect\citebk{[BR 1977]}.

Third, let us consider the scattering process for systems of type
I . Since the potential decays as $|q_{j}-q_{k}|^{-2}\to \infty $,
we have \be q_{j}(t)\sim p_{j}^{\pm }t + q_{j}^{\pm },\quad t\to
\pm \infty .\ee If the particles are numbered in such a way that
$q_{1}<q_{2}<\ldots <q_{n}$, then $p_{1}^{-}>p_{2}^{-}>\ldots
p_{n}^{-}$ and $p_{1}^{+}<p_{2}^{+}<\ldots < p_{n}^{+}$. The
explicit formulae show that the asymptotic values $p_{j}^ {\pm }$
and $q_{j}^{\pm }$ differ only by a permutation: \be p_{1}^{-} =
p_{n}^{+},\quad p_{2}^{-} = p_{n-1}^{+},\ldots ,p_{n}^{-}=
p_{1}^{+}, \ee \be q_{1}^{-} = q_{n}^{+},\quad q_{2}^{-} =
q_{n-1}^{+},\ldots ,q_{n}^{-}= q_{1}^{+}. \ee

A converse statement by Sinai is also true:
\begin{quote}
If we have a one-dimensional $n$-body problem for which the repulsive pair
potential decays as some power of the distance and the asymptotic momenta are
related by (18), then the potential will have the form I.
\end{quote}

The explicit formulae for the solutions show a simple relationship between
the trajectories in the systems of both types I and V.\cite{[Pe 1978]} 
Let $q_{j}
(t)$ be the solution for the system of type I, and $\tilde q_{j}(t)$ the
solution for the system of type V. Then
\be \tilde q_{j}(t) = q_{j}\,\bigl(\omega ^{-1}\,\tan \,(\omega t)\bigr)
\cdot \cos \,\omega t. \ee
Finally, we consider the symmetric polynomials of degree $k$ in the
coordinates $q_{j}(t)$. It follows that these will be polynomials of degree
$k$ in $t$ if $\omega =0$, or in both $\sin \,\omega t$ and $\cos \,\omega
t$ if $\omega \neq 0$.

It is clear that all the above results are valid also for other
systems of types I and V which have the Lax representation,
i.e.~to the general systems I $BC_{n}$ and V $BC_{n}$. For this
purpose, one has to consider the configuration space of $(2n+1)
\times (2n+1)$ traceless Hermitian matrices satisfying \be x f +
fx^{+} = 0,\ee where
\begin{displaymath} f = \left| \begin{array}{ccc} 0&0&J\\
0&1&0\\ J&0&0\end{array} \right| \end{displaymath}
and $J$ is an $n\times n$ matrix with units on the opposite diagonal.
The formulae (13) and (16) involve the matrix $L(0)$ and
\be Q = {\rm diag}\,(q_{1},\ldots ,q_{n},0,-q_{n},\ldots ,-q_{1}).\ee

\subsection{Integration of equations of motion for trigonometric case}
\setcounter{equation}{0}

In this section we construct the solutions of the equations of motion for
systems of types II and III.

Let us discuss first the systems of type II. We consider the space
$X_{n}^{-}$ of $n\times n$ positive-definite Hermitian matrices
with determinant 1. This is the homogeneous space: $X_{n}^{-} =
SL\,(n,{\bf C})\, /\,SU(n)$, \be x\to gxg^{+},\quad x\in
X_{n}^{-}. \ee In particular, any point $x$ can be represented as
\be x = gg^{+}. \ee The space $X_{n}^{-}$  has a metric ${\rm
d}s^{2}$ invariant under transformations of type (1) \be {\rm
d}s^{2} = {\rm tr}\,\Bigl((x^{-1}\,{\rm d}x)^{2}\Bigr). \ee The
curvature of $X_{n}^{-}$  relative to this metric is non-positive,
so we use the minus superscript in our notation. The equation of
geodesics may be easily obtained: \be {{\rm d}\over {{\rm
d}t}}\,\Bigl(x^{-1}(t)\,{{\rm d}\over {\rm d}t} \,x(t)\Bigr) = 0.
\ee

Let us consider $X_{n}^{-}$ as a submanifold of $SL(n,{\bf C})$. The matrices
$x^{-1}(t)\,\dot x(t)$ and $\dot x(t)\,x^{-1}(t)$ are the tangent vectors
to $SL(n,{\bf C})$ at the identity, but are not tangent to $X_{n}^{-}$ in
general. However their sum is tangent to $X_{n}^{-}$. Therefore instead of
(4) we shall consider the equation
\be {{\rm d}\over {\rm d}t}\,\left({{x^{-1}\dot x
+\dot xx^{-1}}\over 2}\right) = 0. \ee
Any geodesic in $X_{n}^{-}$ can be written as
\be x(t) = b\,\exp \,\{2at\}\,b^{+}, \ee
where $b\in SL(n,{\bf C}),\,\,a^{+}=a,\,\,{\rm tr}\,a =0.$

From linear algebra it is known that the Hermitian matrix can be
reduced to diagonal form by a unitary transformation. Then for any
curve $x(t)$ in $X_{n}^{-}$ we have \be x(t) = u(t)\,\exp
\,\{2\gamma Q(t)\}\,u^{+}(t), \ee where $u\in SU(n),\,\gamma $ is
a parameter and \be Q={\rm diag}\,[q_{1},\ldots ,q_{n}]. \ee This
is the so-called spherical coordinate system on $X_{n}^{-}$. With
respect to the parameterization (7) of $x(t)$ we have \be {1\over
2}\,(x^{-1}\dot x+\dot xx^{-1}) = 2\gamma uLu^{+},\ee where
\begin{eqnarray}
L&=&P+{1\over {4\gamma }}\,[{\rm e}^{-2\gamma Q}\,M{\rm e}^{2\gamma Q}-
{\rm e}^{2\gamma Q}\,
M{\rm e}^{-2\gamma Q}], \\
M &=& u^{-1}\dot u\quad \mbox{is the angular velocity},
\end{eqnarray}
and $P=\dot Q$ is the relative velocity in the coordinate system (7).
Differentiating (9) with respect to time we find
\be {{\rm d}\over {\rm d}t}\,\left({{x^{-1}\dot x
+\dot xx^{-1}}\over 2}\right) =
2\gamma u\,(\dot L-[L,M])\,u^{+}. \ee
If $x(t)$ is geodesic, the left-hand side of this equation will vanish. It
follows that the geodesic flow on $X_{n}^{-}$ has the Lax representation
with the matrices $L$ and $M$ given by (10) and (11).

It can be verified directly that the Lax pair for systems of type II
satisfies (10), provided that the function $x(\xi)$ entering in
$L$ is $x(\xi)=\gamma \,\coth \,\gamma \xi $.

Thus if the geodesic $x(t)$ is chosen in such a way that under the
spherical parameterization (7) its angular velocity (11) has
standard form then the evolution of the radial part $Q(t)$ will
have the Lax representation. As was shown before, this is
equivalent to the original Hamiltonian system.

The special choice of geodesics has a simple mechanical interpretation. Let
us consider the ``angular momentum'' $\mu ={1\over 2}\,(x^{-1}\dot x-\dot {x}
x^{-1})$ of the curve $x(t)$ relative to the Hamiltonian action of the group
$SU(n)$ on the phase space. It turns out that if the ``angular velocity''
$M$ has the standard form, the ``angular momentum'' will be time-independent
and will be given by
\be \mu =-2i\gamma ^{2}g\,(v\otimes v-I),\quad v = (1,\ldots ,1).\ee

On the other hand, the ``angular momentum'' of the geodesic (6) is
\be \mu = -(bab^{-1}-(b^{+})^{-1}ab^{+}).\ee
By comparing these two expressions we obtain the constraints for the
parameters $(a,b)$ of the geodesics which are projected onto the
trajectories of the systems of type II. Thus the geodesic flow (5) is
reduced, by the spherical projection at the fixed
 ``angular momentum'' (13), to the Sutherland system.

The above construction yields immediately an explicit expression for the
trajectories. Let us indicate the solution of the Cauchy problem. Setting
$u(0)=I$ we find from (6) and (7)
\be b = \exp \,\{\gamma Q(0)\},\ee
\be a_{jk} = \gamma p_{j}(0)\,\delta _{jk}+{\rm i}\gamma ^{2}g\,
\sinh ^{-1}\,\bigl(\gamma (q_{j}(0)-q_{k}(0))\bigr)\,(1-\delta _{jk}). \ee
So we arrive at the following result: \newline
{\em the quantities $2\gamma q_{j}(t)$ (i.e.~the solutions of the equations
of motion for a system of type II) are the logarithms of the eigenvalues
of the matrix
\be x(t) = b\exp \,\{ 2at\}\,b^{+},\ee
where the matrices $a$ and $b$ are given by (15), (16).}

The transition to the solutions for systems of type III is
performed formally by substituting ${\rm i}\gamma $
 for the parameter $\gamma $
in the above formulae. Geometrically, this substitution represents the
transition from the space $X_{n}^{-}$ of the positive Hermitian matrices
to the space of the unitary matrices $X_{n}^{+} = SU(n)$.
The group $SU(n)$ is a Riemannian symmetric space of nonnegative curvature
with respect to the invariant metric (3). The relationship between
 $X_{n}^{-}$ and $X_{n}^{+}$
is the special case of duality established by E.~Cartan for a large class
of symmetric spaces which generalizes the well-known relationship between
hyperbolic and spherical geometries.

Analogously, the more complicated systems of type III $BC_{n}$
result from geodesic flows on the complex Grassmann manifolds
$SU(2n+1)\,/\,SU(n)\otimes U(n+1)$ while the systems of type II
$BC_{n}$ are related to the dual noncompact symmetric spaces
$SU(n+1,n)\,/\,SU(n)\otimes U(n+1)$. In this case, the solutions
are also obtained by diagonalizing the matrix of form (6) where
$b\in SU(n+1,n)$ and $a$ satisfies the condition \be af+fa^{+} =
0,\ee
\begin{displaymath} f=\left(\begin{array}{ccc} 0&0&J\\ 0&1&0\\ J&0&0
\end{array} \right),\end{displaymath} \nonumber
\begin{displaymath} J=\left(\begin{array}{cccc} 0&\ldots &0&1\\
0&\ldots &1&0\\ \vdots &\ddots &\vdots &\vdots \\ 1&\ldots &0&0\end{array}
\right),\end{displaymath}\nonumber
and
\be Q = {\rm diag}\,[q_{1},\ldots ,q_{n},0,-q_{n},\ldots ,-q_{1}].\ee

\subsection{Systems with two types of particles.}
\setcounter{equation}{0}

A simple change of variables proposed by F. Calogero\,\cite{[Ca 1975]} gives 
the
generalization of the systems of type II. We shall consider the change of
variables
\be q_{j}\to q_{j}+{\rm i}{\pi \over {2\gamma }},\quad 0<n_{1}<j\leq n.\ee
Then the potential
\be U(q) = g^{2}\,\sum _{k<j}\,\gamma ^{2}\,\sinh ^{-2}\,\gamma (q_{j}-
q_{k}) \ee
takes the form:
%\vspace{4cm}
\be
U(q) =g^{2}\,\sum _{j<i\leq n_{1},\atop n_{1}<j<i\leq n}\gamma ^{2}\sinh ^{-2}\,
\gamma (q_{i}-q_{j}) -
g^{2}\,\sum _{i>n_{1},\atop j\leq n_{1}}\,\gamma ^{2}\,\cosh ^{-2}\,\gamma
(q_{i}-q_{j}).
\ee

So we may consider this system as containing $n_{1}$ particles
of the same sign and $n_{2}=n-n_{1}$ particles of the opposite sign. To be
more precise we assume that $n_{1}\geq n_{2}$.
Obviously, the substitution (1) does not affect the main results of the
previous Section. In particular, the quantities $\exp \,2\gamma q_{j}(t)$
are the eigenvalues of the matrix $x(t)$. However, this prescription does
not tell us whether the system with the potential (3) has bound
states or not. Using the geometric approach one can show that the bound states
always exist but are unstable except in the case 
$n_{1}=n_{2}=1$.\,\cite{[OR 1978]}

\subsection{Integration of equations of motion for non-periodic Toda lattice}
\setcounter{equation}{0}

Let us consider first the ordinary non-periodic Toda lattice with
the potential \be U(q) = \sum _{j=1}^{n-1}\,\exp
\,2(q_{j}-q_{j+1}).\ee
Let us remind first that this system was considered first by 
Toda.\cite{[To 1967]}
The Lax representation for standard Toda system
was found by Flaschka\,\cite{[Fl 1974]}
 and Manakov\,\cite{[Ma 1975]} and for general case by 
Bogoyavlensky.\cite{[Bo 1976]}
The explicit integration of equations of motion was achieved in 
Refs.\,\protect\citebk{[OP 1979]}, \protect\citebk{[Ko 1979]} 
and \protect\citebk{[OP 1981b]}. Below we follow 
Ref.\,\protect\citebk{[OP 1979]}.
According to the general idea of the
projection method we should indicate the larger phase space
$\tilde {\cal M}$ whose projection would be the phase space of the
dynamical system with potential (1). In our case, this is the
phase space of the geodesic flow on the space of the symmetric
positive-definite matrices with determinants equal to 1
$$X_{n} = SL(n,{\bf R})\,/\,SO(n).$$
The Riemannian metric and geodesics are given by the same formulae as in the
the case of the Sutherland system (the matrices are real).

We shall now define the projection. Let $Z$ be the subgroup of
$SL(n, {\bf R})$ consisting of the upper triangular
matrices with units on the principal diagonal, and let $H$ be the subgroup
of the diagonal matrices. Any matrix $x$ in $X_{n}$ can be written uniquely
as
\be x = z(x)\,h^{2}(x)\,z^{t}(x),\quad z(x)\in Z,\quad h(x)\in H,\ee
where $z^{t}$ is the matrix transposed to the matrix $z$,
The pair $(h(x),z(x))$ is called the {\em horospherical coordinates} in the
space $X_{n}$. For $n=2$, $X_{2}$ is the Lobachevsky plane and $(h,z)$ are
the standard horospherical coordinates. The diagonal matrix $h(x)={\rm diag}\,
[h_{1},\ldots ,h_{n}]$ is called the horospherical projection. Let $\Delta
_{j}(x)$ be the $j$-th lower principal minor of $x$. Then the horospherical
projection is given by formula
\be h_{1} = {1\over {\Delta _{n-1}}},\quad h_{2} = {\Delta _{n-1}\over
\Delta _{n-2}},\ldots ,h_{n-1} = {\Delta _{2}\over \Delta _{1}},\quad h_{n}=
\Delta _{1}.\ee
We have the two following propositions:
\begin{enumerate}
\item Under the condition
\be \begin{array}{l} 2(bab^{-1})_{j,j-k} = \left\{ \begin{array}{ll}
1,&k=1,\\ 0,&k>1\end{array} \right. \end{array} \ee
the horospherical projection (2) maps the geodesic flow
into the motion defined by the potential (1).
\item Let
$$(q^{0},p^{0}) = (q_{1}^{0},\ldots ,q_{n}^{0},\,p_{1}^{0},\ldots ,
p_{n}^{0})$$ be the initial data for the non-periodic Toda
lattice. We define the $n\times n$ matrix $a$ by \be
a_{jk}=p_{j}^{0}\,\delta _{jk}+e^{q_{j-1}^{0}-q_{j}^{0}}\,\delta _
{j,k+1}+e^{q_{j}^{0}-q_{j+1}^{0}}\,\delta _{j,k-1},\ee and let
$\Delta _{j}$ be the $j$-th lower principal minor of $\exp \, \{
2at\}$. Then the solution of the equations of motion is given by
the formula: \be q_{j}(t) = q_{j}^{0}+{\rm log}\,{{\Delta
_{n+1-j}}\over \Delta _{n-j}}.\ee
\end{enumerate}

To prove this, we consider the horospherical decomposition (2) of the geodesic
$x(t)$:
\be x(t) = z(x)\,{\rm e}^{2Q(t)}\,z^{t}(x).\ee
Then
\be \dot {x}x^{-1} = z\,\{ 2P+e^{2Q}\,M^{t}\,{\rm e}^{-2Q}+M\}\,z^{-1},\ee
where $P=\dot Q$, and
\be M = z^{-1}\dot z.\ee
Let us denote
\be L = P+{1\over 2}\,M+{1\over 2}\,{\rm e}^{2Q}\,M^{t}\,e^{-2Q}.\ee
Then
\be \dot {x}x^{-1} = 2zhz^{-1}.\ee
Differentiating this again we find
\be {{\rm d}\over {\rm d}t}\,\dot x x^{-1}
 = 2z\,\bigl(\dot L -[L,M]\bigr)\,z^{-1}.\ee
Since $x(t)$ is geodesic, the left-hand side of
(12) vanishes. Thus the geodesic flow has the Lax representation with
matrices (9), (10). Note that $M$ lies in the Lie algebra of the group $Z$.

On the other hand, the Lax pair of the Toda lattice also obeys the relation
(10). Moreover, the matrix $M$ is triangular and has a very special form.
This condition singles out the particular class of geodesics.
It can be put in a more natural form by using the moment map $\mu $ for the
action of $Z$:
\be \mu = (\dot x x^{-1})_{-},\ee
where the minus subscript means that all entries lying on and above the
main diagonal of the matrix in the brackets, are replaced by zeros.
Equation (10) shows that the ``angular momentum'' of the geodesics related
to the Toda lattice is
\be \{ \mu \}_{jk} = \delta _{j,k+1}.\ee
Calculating the ``angular momentum'' (13) of the geodesic and using (14)
we obtain (4) which proves the first proposition.

To prove the second proposition we compare the expression
for the geo\-de\-sics
\be x(t) = b\,\exp \,(2at)\,b^{+},\ee
and the horospherical decomposition (7). Taking $z(0)=I$ we find
\be b = \exp \,Q(0) = {\rm diag}\,[e^{q_{1}^{0}},\ldots ,e^{q_{n}^{0}}].\ee
Equation (11) gives the expression (5) for the matrix
$a$ in terms of the initial data. From (15) and (7) we find
\be {\rm e}^{2Q(t)} = z^{-1}b\,\exp \,\{ 2at\}\,b^{t}\,(z^{t})^{-1}.\ee
Since $b$ is the diagonal matrix, we have $z^{-1}b=bz_{1}$ where $z_{1}$ is
some element of $Z$. Therefore (17) can be written as
\be \exp \,2\bigl(Q(t)-Q(0)\bigr) = z_{1}\,\exp \,\{ 2at\}\,z_{1}^{t}.\ee
Combining this with the definition of horospherical projection ((2) and
(3)) we obtain (6).

To conclude, we note that in the case under consideration, all calculations
are reduced to taking the matrix exponent, with a subsequent computation
of minors, i.e.~to pure algebraic operations.

\subsection{Elliptic case}
\setcounter{equation}{0}

Let us remind first that for $N$-body elliptic Calogero system
two Lax representations are known: one without spectral 
parameter\,\cite{[Ca 1975]}
 and another with spectral parameter.\cite{[Kr 1980]} 
This gives the possibility 
 to apply methods of algebraic geometry for integration of equations 
 of motion.\cite{[Kr 1980]}
For this system recently was found the generalization of projection method.
Namely, in Ref.\,\protect\citebk{[GP 1999]} it was proved that every symmetric elliptic
 function in coordinates $q_1, q_2, ..., q_N$ is a meromorphic function
 of time $t$. This gives the possibility to obtain the explicit formulae
 for solution of equation of motion in terms of genus $N-1$ theta functions.
The proof of these statements and other details may be found in 
Ref.\,\protect\citebk{[GP 1999]}.

%\newpage

\section*{Appendix: Completely integrable classical systems connected with semi-simple
Lie algebras. II. Preprint ITEP -- 27 (1976) }
\setcounter{equation}{0}
\addcontentsline{toc}{section} {\numberline{}Appendix:
Completely integrable classical systems connected
with semi-simple Lie algebras. II.   Preprint ITEP -- 27, 1976 }

\medskip
\centerline{\bf Abstract}
\medskip
\begin{quote}
{\em The complete integrability of a class of dynamical systems of
the type considered in the preceding paper}\,\cite{[OP 1975]} {\em but with the
potential $V(q)=q^{-2}+cq^2$ is proved.}
\end{quote}
\setcounter{equation}{0}

\noindent Let us consider the classical dynamical system with $n$
degrees of freedom that is described by the Hamiltonian \be H(p_k,
q_l) = \frac {1}{2}\,\sum _{k=1}^n\,p_k^2+U(q_1,\ldots ,q_n).\ee

The preceding paper by Olshanetsky and Perelomov\,\cite{[OP 1975]} (see also
preceding papers\,\cite{[Mo 1975],[CMR 1975],[Ca 1975]}) considers the systems
of the type (1) with a potential
\be U_0(q_1,\ldots ,q_n) = g^2\,\sum
_{k<l}\,[V(q_k-q_l)+\varepsilon V(q_k+q_l)] +g_1^2\,\sum
_{k=1}^n\,V(q_k)+g_2^2\,\sum _{k=1}^n\,V(2q_k), \ee where
constants $\varepsilon , g, g_1$ and $g_2$ satisfy the condition
\[ \varepsilon =0\,\,\mbox{or\,\,1\,\,for}\,\,g_1=g_2=0;\quad \varepsilon =1,
\,\,\mbox{for}\,\,g_1^2+g_2^2>0
\]
and the potential $V(q)$ has one of the following forms
\begin{eqnarray}
&&{\rm \bf I.}  \qquad\qquad ~~~V(q)=q^{-2}\,,\nonumber\\
&&{\rm \bf II.}  \qquad\qquad ~V(q)=a^2\,\sinh ^{-2}a q\,,\nonumber\\
&&{\rm \bf III.}  \qquad\qquad V(q)=a^2\,\sin ^{-2}a q\,,\nonumber\\
&&{\rm \bf IV.}  \qquad\qquad V(q)=a^2\,{\wp }(a q)
\end{eqnarray}
(${\wp}(q)$ is the Weierstrass function). In  
Ref.\,\protect\citebk{[OP 1975]} it was
also shown that these systems are related to the semi-simple Lie
algebras.

The aim of the present paper is to extend results of 
Ref.\,\protect\citebk{[OP 1975]} to
the systems of the type (1) with the potential (2), where
$V(q)=q^{-2}+cq^2$.\footnote{\,After this paper has been performed
we got the possibility to acquaint with Adler's preprint\,\cite{[Ad 1975]} in
which the similar results were obtained.} Here, besides the
integrals of motion $I_k$, we obtain also the explicit expressions
for quantities $B_k^*$ and $B_k$ which are a classical analog of
quantum operators of creation and annihilation $B_k^+$ and $B_k$
introduced in Ref.\,\protect\citebk{[Pe 1971]}. Note that in 
Ref.\,\protect\citebk{[Pe 1971]} the algorithm
 was given for finding of  the operators $B_k^+$ and $B_k$, and the
explicit form of the first three of them was 
found.\footnote{\,Note that analogously to Ref.\,\protect\citebkcap{[Ca 1975]} 
it may be shown
that in the quantum case ($p_k\to -i\,{\partial }/ {\partial
q_k}$) operators $B_k$ and $B_k^+$ are well-defined. Thereby, we
get for them the explicit expressions.}

Note first of all that under the fulfillment of condition $\sum
_{j=1}^n\, q_j = 0$ the system of interest described by the
Hamiltonian (1) with the potential (2) and $V(q)=q^{-2}+\alpha
^2q^2$ is equivalent to the system with the potential \be
U=U_0+\frac{\omega ^2}{2}\,\sum _{k=1}^n\,q_k^2, \ee where $U_0$
is given by the formula (2) with $V(q)=q^{-2}$.

In order to prove the complete integrability of the system under
consideration and to find explicit form of $B_k, B_k^*$ and $I_k$
we generalize the Moser method\,\cite{[Mo 1975]} which in turn applies the Lax
trick.\,\cite{[La 1968]} Namely, we will show that the Hamilton equations \be
\dot q_k = \frac{\partial H}{\partial p_k}=p_k,\,\,\dot p_k=-\frac
{\partial H}{\partial q_k} \ee are equivalent to the matrix
equation \be \frac{d\tilde L}{dt}=[\tilde L, M]-i\omega \tilde L,
\ee where matrices $M$ and $L$ depend on $p_k$ and $q_l$, and the
matrix $iM$ is Hermitian.

Let us first of all note that as it is shown in 
Refs.\,\protect\citebk{[OP 1975]},
\protect\citebk{[Mo 1975]}, \protect\citebk{[CMR 1975]} and 
\protect\citebk{[Ca 1975]}
for the system described by the Hamiltonian $H=\frac{1}{2}\,\sum
_{k=1}^n\,p_k^2+U_0$, where $V(q)$ is given by the formula (3),
the Hamilton equations are equivalent to the matrix equation \be
\frac{dL}{dt}=[L,M] \ee (the explicit form for the matrices
$iM=(-D+Y)$ and $L=P+iX$ is given in the above mentioned papers).
In the case of interest we take \be \tilde L = L-i\,\omega Q,\ee
where the matrix $Q$ is obtained from the matrix $P$ replacing
$p_k$ by $q_k$ and correspondingly the matrix $\tilde L$ is
obtained from the matrix $L$ replacing $p_k$ by $b_k=p_k-i\,\omega
q_k$.

Then, as it can be easily checked, equations (6) are a consequence
of the equation (8) and of identities \be \frac{d\tilde L}{dt}
=[L,M]-i\omega (P-i\omega Q),\qquad L=P+[M,Q]. \ee The check of
the first of these identities is very simple while for the check
of the second one it is necessary to make use of explicit
expressions for the matrices $L$ and $M$ from 
Refs.\,\protect\citebk{[OP 1975]}, \protect\citebk{[Mo 1975]}, 
\protect\citebk{[CMR 1975]} and \protect\citebk{[Ca 1975]}.

Let us now turn to obtaining of consequences from the equation
(6). Form the quantities 
\be 
B_k=\mbox{tr}\,(\tilde
L^k)\,\quad \mbox{and}\,\quad B_k^*=\mbox{tr}\,\left( \left( \tilde
L^+\right) ^k\right) . \ee Then, as it follows from (6), \be \dot
B_k(t) = -\,ik\omega \,B_k(t),\qquad B_k(t)=\exp \,(-\,ik\omega
t)\, B_k(0). \ee Thus, the quantities $B_k(t)$, not being the
integrals of motion, possess, however, a very simple time
dependence.\footnote{\,The quantities $B_k(t)$ completely
determine the evolution of considered system. They are rational
functions of variables $p_k$ and $q_l$. Expressing the coordinates
and momenta through $B_k(t)$ and $B_l^*(t)$ we get an explicit
expression for $p_k(t)$ and $q_l(t)$ and thereby completely
integrate the system of Hamilton's equations~(5).}

Let us now find the integrals of motion. It is easy to see that
these are both $B_k^*B_k$ and $\{ B_k,B_k^*\}$. Thus the knowledge
of quantities $B_k$ is sufficient to find the invariants. It will
be more convenient for us to use another set of invariants. Let us
consider matrices $N_1=\tilde L^+ \tilde L$ and $N_2=\tilde
L\tilde L^+$. It follows from (6) that they satisfy the equation
\be \frac {dN_k}{dt}=[N_k,M],\qquad k=1,2. \ee Consequently, the
matrix \be N=\frac{1}{2}\left( \tilde L^+\tilde L+\tilde L\tilde
L^+\right) =L^2+\omega ^2Q^2 \ee also satisfies the equation \be
\frac{dN}{dt}=[N,M]. \ee Therefore, the eigenvalues of matrix $N$
do not change with time and consequently the quantities \be
I_k=\mbox{tr}\,(N^k) \ee are the integrals of motion.

In conclusion, let us prove that the quantities $B_k$ are in
involution, i.e., $\{B_k, B_l\}=0$. Here
\[ \{A, B\}=\sum _{k=1}^n\,\left( \frac{\partial A}{\partial p_k}\,
\frac{\partial B}{\partial q_k}-\frac{\partial A}{\partial q_k}
\,\frac{\partial B}{\partial p_k}\right )\] is the Poisson bracket
for $A$ and $B$.

Indeed, the quantities $B_k$ are obtained from integrals of motion
$I_k^{(0)}$ for the system described by Hamiltonian $H_0$
replacing in them $p_k$ by $b_k=p_k-i\,\omega q_k$. With this
replacement
\[
\frac{\partial I_k^{(0)}}{\partial p_j}\to \frac{\partial
B_k}{\partial b_j}, \qquad \frac{\partial I_k^{(0)}}{\partial
q_j}\to \frac{\partial B_k} {\partial q_j}-i\omega \,\frac
{\partial B_k}{\partial b_j}.\] Thus, \be \{B_k, B_l\}=\sum
_j\,\left( \frac{\partial B_k}{\partial p_j}\,\frac {\partial
B_l}{\partial q_j}-\frac{\partial B_k}{\partial q_j}\,
\frac{\partial B_l}{\partial p_j}\right) = \sum _j\,\left(
\frac{\partial B_k} {\partial b_j}\,\frac {\partial B_l}{\partial
q_j}-\frac{\partial B_k} {\partial q_j}\,\frac{\partial
B_l}{\partial b_j}\right) = 0, \ee since as it follows from 
Refs.\,\protect\citebk{[OP 1975]}, \protect\citebk{[Mo 1975]}, 
\protect\citebk{[CMR 1975]} and \protect\citebk{[Ca 1975]},
\[ \{I_k^{(0)}, I_l^{(0)}\} = 0.\]

\newpage
\section*{References}
\addcontentsline{toc}{section} {\numberline{}References}

\def\b{\hfill\break}

\begin{thebibliography}{99}

\bibitem
{[AG 1996]}
%{AG} 
Alekseevsky D. and Guha P.
``On decomposability of Nambu-Poisson tensor,'' {\em Acta Math.\
Univ.\ Comenianae}, {\bf 65}, No.~1,\, 1-9 (1996).

\bibitem{[AH 1988]} Atiyah M.\ and Hitchin N.\
         {\em The Geometry and Dynamics of Magnetic Monopoles},
 (Princeton Univ.\ Press, Princeton, 1988).

\bibitem{[Ad 1975]} Adler M., Courant Institute Preprint (1975).

\bibitem{[Ar 1989]} Arnold V.I.\
              {\em Mathematical Methods of Classical Mechanics},
Graduate Texts in Mathematics, No.~60, Second Edition,
( Springer, Berlin, Heidelberg, New York, 1989).

\bibitem{[APP 1996a]} Azc\'arraga J.A., Perelomov A.M.\ and
 P\'erez Bueno J.C.\
``New generalized Poisson structures,''
 {\em  J.\ Phys.}\ A {\bf 29},  L151-L157 (1996).

\bibitem{[APP 1996b]} Azc\'arraga J.A., Perelomov A.M.\ and
 P\'erez Bueno J.C.\
``The Schouten-Nijenhuis bracket, cohomology and generalized
Poisson structures,''
{\em J.\ Phys.}\ A {\bf 29}, 7993-8010 (1996).

\bibitem{[AIPP 1977]} Azc\'arraga J.A., Izquierdo J.M., Perelomov A.M.
and P\'erez Bueno J.C.\
``The $Z_2$-graded Schouten-Nijenhuis bracket and generalized
     super-Poisson structures,''
{\em J.\ Math.\ Phys.}\ {\bf 38}, 3735-3749 (1977).

\bibitem{[BR 1977]} Barucchi G.\ and Regge T.\
        ``Conformal properties of a class of exactly solvable $N$-body
problems in space dimension one,''
{\em J.\ Math.\ Phys.}\ {\bf 18}, 1149--1153 (1977).

\bibitem{[Bo 1976]} Bogoyavlensky O.I.\
              ``On perturbations of the periodic Toda lattices,''
 {\em   Comm.\ Math.\ Phys.}\ {\bf 51}, 201--209 (1976).

\bibitem{[Bou 1855]} Bour J.\
        ``Sur l'int\'egration des \'equations diff\'erentielles de la
m\'ecanique analytique,'' 
{\em J.\ Math.\ Pures Appl.}\ {\bf 20}, 185--200 (1855).

\bibitem{[CIMP 1994]} Cari\~ nena J., Ibort A., Marmo G.\ and
         Perelomov A.\
         ``On the geometry of Lie algebras and Poisson tensors,''
{\em J.\ Phys.}\ A {\bf 27}, 7425--7449 (1994).

\bibitem{[Ca 1971]} Calogero F.\
         ``Solution of the one-dimensional $n$-body problem with quadratic
and/or inversely quadratic pair potentials,''
{\em J.\ Math.\ Phys.}\ {\bf 12}, 419--436 (1971).

\bibitem{[CMR 1975]} Calogero F., Marchioro C.\ and Ragnisco O.\
        ``Exact solution of the classical and quantal one-dimensional
many-body problems with the two-body potential
$V_{a}(x)=g^{2}a^{2}\, /\,\sinh ^{2}(ax)$,'' {\em Lett.\ Nuovo Cim.}\ 
{\bf 13}, 383-387 (1975).

\bibitem{[Ca 1975]} Calogero F.\
        ``Exactly solvable one-dimensional many-body systems,''
       {\em  Lett.\ Nuovo Cimento}, {\bf 13}, 411--416 (1975).

\bibitem{[Ca 1976]} Calogero F.\
        ``On a functional equation connected with integrable many-body
problems,'' {\em Lett.\ Nuovo Cim.}\ {\bf 16}, 77--80 (1976).

\bibitem{[Ca 2001]} Calogero F.\
 {\em Classical many-body problems in one-, two- and three-dimensional space
 amenable to exact treatments
 (solvable and/or integrable and/or linearizable)},
 (Springer-Verlag, 2001).

\bibitem{[Fl 1974]} Flaschka H.\
        ``The Toda lattice. Existence of integrals,''
                  {\em  Phys.\ Rev.}\ B {\bf 9}, 1924-1925 (1974).

\bibitem{[Go 2001]} Goriely A.\
{\em Integrability and non-integrability of dynamical systems},
(World Sci., Singapore, 2001).

\bibitem{[GGKM 1967]} Gardner C., Greene J., Kruskal M.\ and Miura R.\
``Method for solving the Korteweg-de Vries equation,''
{\em Phys.\ Rev.\ Lett.}\ {\bf 19}, 1921--1923 (1967).

\bibitem{[GM-ORSW 1990]} Grammaticos B., Moulin-Ollagnier J., Ramani A.,
        Strelcyn J.-M.\ and Wojciechowski S.\
''Integrals of quadratic ordinary differential equations in
$R^3$ : The Lotka-Volterra system,''
      {\em  Physica}, A {\bf 163}, 683 (1990). 

\bibitem{[GMP 1993]} Grabowski J., Marmo G.\ and Perelomov A.\
       ''Poisson structures: towards a classification,''
        {\em Mod.\ Phys.\ Lett.}\ A {\bf 18}, 1719-1733 (993).

\bibitem{[GN 1993]} G\"umral H.\ and Nutku Y.\
       ''Poisson structure of dynamical systems with three degree of
        freedom,'' 
     {\em   J.\ Math.\ Phys.}\ {\bf 34}, No.~12, 5691-5723 (1993).

\bibitem{[GP 1999]} Gavrilov L.\ and Perelomov A.M.\
 ``On the explicit solutions of the elliptic Calogero system,''
        {\em J.\ Math.\ Phys.}\ {\bf 40}, 6339--6352 (1999).

\bibitem{[Ha 1881]} Halphen G.-H.\
``Sur un syst\`eme d'\'equations diff\'erentielles,''
                 {\em Compt.\ Rend.}\ {\bf 92}, 1101--1103 (1881).

\bibitem{[Ja 1866]} Jacobi C.\
 ``Problema trium corporum mutuis attractionibus cubis
 distantiarum inverse proportionalibus recta linea se moventium,'' 
 In: {\em Gesammelte Werke}, (Berlin, 1866), Vol.~4, pp.\, 533--539.

\bibitem{[KKS 1978]} Kazhdan D., Kostant B.\ and
Sternberg S.\ 
``Hamiltonian group actions  and dynamical
systems  of Calogero type,'' {\em Commun.\ Pure Appl.\ Math.}\ {\bf 31},
481--508 (1978).

\bibitem{[KK 1927]} Kermack W.\ and McKendrick A.\
 ``Contribution to the mathematical theory of epidemics. I.''
 {\em Proc.\ Roy.\ Soc.\ London}, A {\bf 115}, 700 (1927).

\bibitem{[KK 1932]} Kermack W.\ and McKendrick A.\
``Contribution to the mathematical theory of epidemics. II. The
problem of endemicity,''
 {\em Proc.\ Roy.\ Soc.\ London}, A {\bf 138}, 55--83 (1932).

\bibitem{[KK 1933]} Kermack W.\ and McKendrick A.\
``Contribution to the mathematical theory of epidemics. III.
Further study of the problem of endemicity,''
 {\em Proc.\ Roy.\ Soc.\ London},
 A {\bf 141}, 94--122 (1933).

\bibitem{[Ko 1979]} Kostant B.\
``The solution to a generalized Toda lattice and representation
theory,''   {\em Adv.\ in Math.}\ {\bf 39}, 195--338 (1979).

\bibitem{[Kr 1980]} Krichever I.M.\
``Elliptic solutions of the Kadomtsev-Petviashvili
              equation and many-body problems,''
              {\em Funct.\ Anal.\ Appl.}\ {\bf 14}, No.~4, 282--290 (1980).

\bibitem{[La 1808]} Lagrange J.-L.\
        {\em M\'emoire sur la th\'eorie des variations des \'el\'ements des
plan\'etes et en particulier des variations des grands axes de
leur orbits}, Mem.\ Cl.\ Sci.\ Math.\ Phys.\ Inst.\ France, 1--72 (1808).

\bibitem{[La 1809]} Lagrange J.-L.\
        {\em Second m\'emoire sur la th\'eorie de la variation des constants
arbitraires dans les probl\'emes de m\'echanique}, Mem.\ Cl.\ Sci.\
Math.\ Phys.\ Inst.\ France, 343--352 (1809).

\bibitem{[La 1968]} Lax P.\
``Integrals of nonlinear equations of evolution
              and solitary waves,''
            {\em  Commun.\ Pure Appl.\ Math.}\ {\bf 21}, 467--490 (1968).

\bibitem{[Lie 1874/75]} Lie S.\
``Begr\"undung einer Invariantentheorie der Ber\"uhrungs
                  Transformationen,''
                  {\em Math.\ Ann.}\ {\bf 8}, 214--303 (1874/75).

\bibitem{[Lio 1855]} Liouville J.\
``Note sur les \'equations de la dynamique,''
{\em J.\ Math.\ Pures Appl.}\ {\bf 20}, 137--138 (1855).

\bibitem{[Lo 1963]} Lorenz E.N.\
``Deterministic nonperiodic flow,''
 {\em J.\ Atmos.\ Sci.}\ {\bf 20}, 130--141(1963).

\bibitem{[Lo 1920]} Lotka A.J.\
``Analytic note on certain rhytmic relations in organic
systems,''
{\em Proc.\ Nat.\ Acad.\ Sci.\ USA}, {\bf 6}, 410--415 (1920).

\bibitem{[MW 1974]} Marsden J. and Weinstein A.\
``Reduction of symplectic manifolds with symmetry,''
{\em Reps.\ Math.\ Phys.}\ {\bf 5}, 121--130 (1974).

\bibitem{[Ma 1975]}  Manakov S.V.\
``Complete integrability and stochastization of discrete
dynamical systems,''
{\em Sov.\ Phys.\ JETP}, {\bf 40}, 269--274 (1975).

\bibitem{[Mo 1975]} Moser J.\
``Three integrable Hamiltonian systems connected with
isospectral deformations,''
{\em Adv.\ Math.}\ {\bf 16}, 197--220 (1975).

\bibitem{[Mo 1980]} Moser J.\
``Various aspects of integrable Hamiltonian systems,''
In: {\em Progress in Mathematics}, {\bf 8}, Dynamical Systems,
 (Birkhauser, 1980) 233--289.

\bibitem{[Na 1973]} Nambu, Y.\
``Generalized Hamiltonian mechanics,''
{\em Phys.\ Rev.}\ D {\bf 7}, 2405--2412 (1973).

\bibitem{[OP 1975]} Olshanetsky M.A.\ and Perelomov A.M.,
 Preprint ITEP No.103 (1975).

\bibitem{[OP 1976a]} Olshanetsky M.\ and Perelomov A.\
 ``Completely integrable Hamiltonian systems connected with
 semisimple Lie algebras,'' {\em Invent.\ Math.}\ {\bf 37}, 93--108 (1976).

\bibitem{[OP 1976b]} Olshanetsky M.\ and Perelomov A.\
``Explicit solution of the Calogero model in the classical case
 and geodesic flows on symmetric spaces of zero curvature,''
{\em Lett.\ Nuovo Cimento}, {\bf 16}, 333--339 (1976).

\bibitem{[OP 1976c]} Olshanetsky M.\ and Perelomov A.\
``Explicit solutions of some completely integrable systems,''
{\em Lett.\ Nuovo Cimento}, {\bf 17}, 97--101 (1976).

\bibitem{[OP 1979]} Olshanetsky M.\ and Perelomov A.,\
``Explicit solutions of classical generalized Toda models,''
{\em Invent.\ Math.}\ {\bf 54}, 261--269 (1979).

\bibitem{[OP 1981a]} Olshanetsky M.\ and Perelomov A.\
``Classical integrable finite-dimensional systems related
 to Lie algebras,''
{\em Phys.\ Reps.}\ {\bf 71}, no.~5, 313--400 (1981).

\bibitem{[OP 1981b]} Olshanetsky M.\ and Perelomov A.\
``Toda lattice as a reduced system,''
{\em Theor.\ Math.\ Phys.}\ {\bf 45}, 843--854 (1981).

\bibitem{[OR 1978]} Olshanetsky M.\ and Rogov  V.\
``Bound states in completely integrable systems with two types
of particles,'' {\em Ann.\ Inst.\ H.\ Poincar\'e}, {\bf A29}, 169--177 (1978).

\bibitem{[PS 1976]} Pydkuyko S.\ and Stepin A.\
``On solution of a differential-functional
equation,'' {\em Funct.\ Anal.\ Appl.}\ {\bf 10}, 155--156 (1976).

\bibitem{[Pe 1971]} Perelomov A.M.\
 ``Algebraical approach to the solution
 of one-dimensional model of $N$ interacting particles,''
 {\em Theor.\ Math.\ Phys.}\ {\bf 6}, 263--283 (1971).

\bibitem{[Pe 1976]} Perelomov A.M.\
 {\em Completely integrable classical systems
 connected with semisimple Lie algebras.}\, II,
 Preprint ITEP-27, 1-7 (1976) [math-ph/0111018].

\bibitem{[Pe 1977]} Perelomov A.M.\
``Completely integrable classical systems connected with
semisimple Lie algebras. III,''
{\em Lett.\ Math.\ Phys.}\ {\bf 1}, 531--534 (1977).

\bibitem{[Pe 1978]} Perelomov A.M.\
``The simple relation between certain dynamical systems,''
{\em Commun.\ Math.\ Phys.}\ {\bf 63}, 9--11 (1978).

\bibitem{[Pe 1990]} Perelomov A.M.\
{\em Integrable Systems of Classical Mechanics and Lie Algebras.}
{\bf I},  (Birkh\"auser, Basel, 1990).

\bibitem{[Pe 1992]} Perelomov A.M.\
``The projection method in classical mechanic,''
 In: {\em Proceedings of the 20 International Conference on Differential
 Geometric Methods in Theoretical Physics}, {\bf I},
 New York City, June 3-7, 1991; (World Sci., Singapore, 1992) 594-602.

\bibitem{[Pe 1995]} Perelomov, A.M.\
{\em Selected topics on classical integrable systems}, Lectures
 at Ecole Polytechnique F\'ed\'eral de Lausanne.

\bibitem{[Po 1809]} Poisson S.\
``M\'emoire sur la variation des constantes arbitraries dans les
 questions de m\'echanique,''
{\em J.\ Ecole Polytec.}\ {\bf 8}, 266--344 (1809).

\bibitem{[Sk 1982]} Sklyanin E.K.\
``Algebraic structures connected with
the Yang--Baxter equation,''
{\em Funct.\ Anal.\ Appl.}\ {\bf 16}, 263--270 (1982).

\bibitem{[Su 1971]} Sutherland B.\
``Exact results for a quantum many-body problem
in one dimension,''
{\em Phys.\ Rev.}\ A {\bf 4}, 2019--2021 (1971).

\bibitem{[Ta 1994]} Takhtajan L.\
``On foundation of generalized Nambu mechanics,''
{\em Commun.\ Math.\ Phys.}\ {\bf 160}, 295--315 (1994).

\bibitem{[To 1967]} Toda M.\
``Vibration of a chain with nonlinear interaction,''
{\em J.\ Phys.\ Soc.\ Japan}, {\bf 20}, 431--436 (1967).

\bibitem{[Vo 1931]} Volterra V.\
{\em Le\c{c}ons sur la Th\'eorie Math\'ematique de la Lutte
pour la Vie}, (Gauthier-Villars, Paris, 1931).

\bibitem{[Wh 1937]} Whittaker E.T.\
{\em A Treatise on the Analytical Dynamics
 of Particles and Rigid Bodies},
 4-th Ed.\ (Cambridge Univ.\ Press, Cambridge, 1937).

\end{thebibliography}
\end{document}
