Oliynyk O. Nonlinear differential equations and systems of linear model representation of non-selfadjoint operators

System of non-linear differential equations of Lax type, on solutions of which a construction of triangular models for commutative systems of linear non-selfadjoint bounded operators is based, is studied. All the solutions of this system in two-dimensional case are obtained. A step-by-step method of construction of all solutions of Lax system based on isospectral transforms is introduced. General form of solutions in the case of linear dependency of the weight matrix-function on solutions is found. It is proved that in the case of polynomial dependency of a weight matrix-function on solutions for r=3,4 some solutions are written in terms of elliptic functions.