Geodesic
Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms
Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 1, pp. 9-23

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A method of classifying exactly and completely integrable emb.eddings in Riemannian or non-Riemannian enveloping Spaces is proposed. It is based on the algebraic approach [6, 8] to the integration of nonlinear dynamical systems. The grading conditions and the spectral composition of the Lax operators, which take values in a graded Lie algebra and distinguish the integrable classes of two-dimensional systems, are formulated in terms of the structure of the tensors of the third fundamental forms. In the framework of the method, each embedding of the three-dimensional subalgebra $\text{sl}(2)$ in a simple finite-dimensional (infinite-dimensional of finite growth) Lie algebra is associated with a definite class of exactly (completely) integrable embeddings of a two-dimensional manifold in a corresponding enveloping space equipped with the structure of . 
@article{TMF_1984_60_1_a1,
     author = {M. V. Saveliev},
     title = {Classification of exactly integrable embeddings of two-dimensional manifolds. {The} coefficients of the third fundamental forms},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {9--23},
     publisher = {mathdoc},
     volume = {60},
     number = {1},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1984_60_1_a1/}
}
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M. V. Saveliev. Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms. Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 1, pp. 9-23. http://geodesic.mathdoc.fr/item/TMF_1984_60_1_a1/