Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms

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 .