Partial differential equations with finite-dimensional solution manifolds

In the paper, overdetermined systems of nonlinear partial differential equations with two independent variables and an arbitrary number of unknown functions are considered. An efficient criterion of finite dimensionality of the solution manifold for a certain class of such systems is obtained. An upper bound of the solution manifold dimensionality of the Bezout type is presented, and algebraic conditions ensuring that this bound is achieved are found. Based on this, a method for constructing the matrix Darboux transform of systems of linear second-order equations is suggested.