\key AckHer \book Ackerman, M., Hermann, R.; Sophus Lie's 1880 Transformation Group Paper; Math Sci Press, Brookline, Mass., 1975\par \key AckHerdi \book Ackerman, M., Hermann, R.; Sophus Lie's 1884 Differential Invariant Paper; Math Sci Press, Brookline, Mass., 1976\par \key Ahlfors \book Ahlfors, L.V.; Complex Analysis; Second Edition, McGraw--Hill Book Co., New York, 1966\par \key BOpol \paper Berchenko, I.A., Olver, P.J.; Symmetries of polynomials; J. Symb. Comp.; a(2000) \par \key BuddIserles \paper Budd, C.J., Iserles, A.; Geometric integration: numerical solution of differential equations on manifolds; Phil. Trans. Roy. Soc. London A; 357 (1999) 945--956\par \key COSTH \paper Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.; Differential and numerically invariant signature curves applied to object recognition; Int. J. Computer Vision; 26 (1998) 107--135\par \key COT \paper Calabi, E., Olver, P.J., Tannenbaum, A.; Affine geometry, curve flows, and invariant numerical approximations; Adv. in Math.; 124 (1996) 154--196\par \key Cartanrm \book Cartan, \'E.; La M\'ethode du Rep\`ere Mobile, la Th\'eorie des Groupes Continus, et les Espaces G\'en\'eralis\'es; Expos\'es de G\'eom\'etrie No. 5, Hermann, Paris, 1935\par \key Cartang \book Cartan, \'E.; La Th\'eorie des Groupes Finis et Continus et la G\'eom\'etrie Diff\'erentielle Trait\'ees par la M\'ethode du Rep\`ere Mobile; Cahiers Scientifiques, Vol. 18, Gauthier--Villars, Paris, 1937\par \key Cartanequiv \inbook Cartan, \'E.; Les probl\`emes d'\'equivalence; Oeuvres Compl\`etes; Part. II, Vol. 2, Gauthier--Villars, Paris, 1953, pp. 1311--1334\par \key Dorodfd \paper Dorodnitsyn, V.A.; Finite difference models entirely inheriting continuous symmetry of original differential equations; Int. J. Mod. Phys. C; 5 (1994) 723--734\par \key FOmcI \paper Fels, M., Olver, P.J.; Moving coframes. I. A practical algorithm; Acta Appl. Math.; 51 (1998) 161--213\par \key FOmcII \paper Fels, M., Olver, P.J.; Moving coframes. II. Regularization and theoretical foundations; Acta Appl. Math.; 55 (1999) 127--208\par \key FOmfmc \preprint Fels, M., Olver, P.J.; Moving frames and moving coframes; University of Minnesota, 1997\par \key GSS \book Golubitsky, M., Stewart, I., Schaeffer, D.G.; Singularities and Groups in Bifurcation Theory; Vol. 2, Springer--Verlag, New York, 1988\par \key Griffithsmf \paper Griffiths, P.A.; On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry; Duke Math. J.; 41 (1974) 775--814\par \key GH \book Griffiths, P.A., Harris, J.; Principles of Algebraic Geometry; John Wiley \& Sons, New York, 1978\par \key GJ \book Griffiths, P.A., Jensen, G.R.; Differential Systems and Isometric Embeddings; Annals of Math. Studies, No. 114, Princeton Univ. Press, Princeton, N.J., 1987\par \key Gug \book Guggenheimer, H.W.; Differential Geometry; McGraw--Hill, New York, 1963\par \key Jensen \book Jensen, G.R.; Higher Order Contact of Submanifolds of Homogeneous Spaces; Lecture Notes in Math., {\bf 610}, Springer--Verlag, New York, 1977\par \key KMO \paper Kamran, N., Milson, R., Olver, P.J. ; Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems; Adv. in Math.; a(2000) \par \key Killingi \paper Killing, W.; Erweiterung der Begriffes der Invarianten von Transformationgruppen; Math. Ann.; 35 (1890) 423--432\par \key Kolchin \book Kolchin, E.R.; Differential Algebra and Algebraic Groups; Academic Press, New York, 1973\par \key Lietg \paper Lie, S.; Theorie der Transformationsgruppen I; Math. Ann.; 16 (1880) 441--528; also {\it Gesammelte Abhandlungen}, Vol. 6, B.G. Teubner, Leipzig, 1927, pp. 1--94; see \rf{AckHer} for an English translation\par \key Liedi \paper Lie, S.; \"Uber Differentialinvarianten; Math. Ann.; 24 (1884) 537--578; also {\it Gesammelte Abhandlungen}, Vol. 6, B.G. Teubner, Leipzig, 1927, pp. 95--138; see \rf{AckHerdi} for an English translation\par \key Lieode \paper Lie, S.; Klassifikation und Integration von gew\"ohnlichen Differentialgleichungen zwischen $x$, $y$, die eine Gruppe von Transformationen gestatten I, II; Math. Ann.; 32 (1888) 213--281; also {\it Gesammelte Abhandlungen}, Vol. 5, B.G. Teubner, Leipzig, 1924, pp. 240--310\par \key Milne \book Milne--Thompson, L.M.; The Calculus of Finite Differences; Macmilland and Co., Ltd., London, 1951\par \key MPVO \paper Moons, T., Pauwels, E., Van Gool, L., Oosterlinck, A.; Foundations of semi-differential invariants; Int. J. Comput. Vision; 14 (1995) 25--48\par \key Os \paper Olver, P.J.; Symmetry groups and group invariant solutions of partial differential equations; J. Diff. Geom.; 14 (1979) 497--542\par \key O \book Olver, P.J.; Applications of Lie Groups to Differential Equations; Second Edition, Graduate Texts in Mathematics, vol. 107, Springer--Verlag, New York, 1993\par \key E \book Olver, P.J.; Equivalence, Invariants, and Symmetry; Cambridge University Press, Cambridge, 1995\par \key Ops \paper Olver, P.J.; Pseudo-stabilization of prolonged group actions. I. The order zero case; J. Nonlinear Math. Phys.; 4 (1997) 271--277\par \key I \book Olver, P.J.; Classical Invariant Theory; Cambridge University Press, Cambridge, 1999\par \key Oji \preprint Olver, P.J.; Joint invariant signatures; University of Minnesota, 1999\par \key Onums \preprint Olver, P.J.; Geometric foundations of numerical algorithms and symmetry; University of Minnesota, 1999\par \key Ov \book Ovsiannikov, L.V.; Group Analysis of Differential Equations; Academic Press, New York, 1982\par \key Juhap \personal Pohjanpelto, J.; 1998\par \key Vara \book Varadarajan, V.S.; Lie Groups, Lie Algebras, and their Representations; Springer--Verlag, New York, 1984\par