Geodesic
Reduction of differential equations with symmetries
Izvestiya. Mathematics, Tome 17 (1981) no. 1, pp. 73-86 
E. M. Vorob'ev. Reduction of differential equations with symmetries. Izvestiya. Mathematics, Tome 17 (1981) no. 1, pp. 73-86. http://geodesic.mathdoc.fr/item/IM2_1981_17_1_a2/
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     title = {Reduction of differential equations with symmetries},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A method for constructing group-invariant solutions of differential equations is described. At the foundation of the method lies a reduction of the dimension of the base of a bundle of $k$-jets of functions $J^k(M^n,R^1)$ by means of a passage to the manifolds of orbits of the contact action of the Lie group of partial symmetries of the differential equation. Only the orbits of a certain submanifold of $J^k(M^n,R^1)$ are considered, an extension of an involutive system of first-order differential equations associated with the action of the group. Bibliography: 7 titles.

[1] Marsden J., Weinstein A., “Reduction of symplectic manyfolds with symmetry”, Rep. Math. Phys., 5:1 (1974), 121–130 | DOI | MR | Zbl

[2] Goldschmidt H., “Integrability criteria for systems of nonlinear differential equations”, J. Diff. Geometry, 1 (1967), 269–307 | MR | Zbl

[3] Vinogradov A. M., “Mnogoznachnye resheniya i printsip klassifikatsii nelineinykh differentsialnykh uravnenii”, Dokl. AN SSSR, 210:1 (1973), 11–14 | Zbl

[4] Dhooghe P. D., “Les transformations de contact sur un espace fibré de jets d'applications”, Compt. rend. Acad. sci. Paris Sér. A–B, 287:16 (1978), A1125–A1128 | MR

[5] Ibragimov N. H., Anderson R. L., “Lie–Backlund tangent transformations”, J. Math. Anal. Applic., 59:1 (1977), 145–162 | DOI | MR | Zbl

[6] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[7] Weinstein A., “Symplectic manifolds and their Lagrangian submanifolds”, Adv. Math., 6 (1971), 329–346 | DOI | MR | Zbl