Two-dimensional nonlinear equations of the string type and their complete integration

The general algebraic construction of the authors [1] is used in an investigation of two-dimensional nonlinear dynamical systems associated by a Lax type representation with the local part of an arbitrary graded Lie algebra. These systems contain as simplest case [for simple finite- and infinite-dimensional (of finite growth) Lie algebras] the generalized two-dimensionalized Toda chain. General (in the sense of the Goursat problem) solutions of the equations characterized by the necessary number of arbitrary functions are constructed. Their structure is illustrated by a detailed consideration of “string” type systems associated with a definite embedding of a three-dimensional subalgebra in the algebra $B_n$. The geometrical interpretation of the corresponding equations is discussed.