Asymptotics of the solution of a system of equations in partial derivatives of the first order with a small parameter at some derivatives

A system of two singularly perturbed equations in partial derivatives of the first order on a plane with a nonlinear right side is considered. A small parameter enters the equations as a multiplier at partial derivatives on different arguments. The theorem of existence of the classical solution of this system with additional conditions on axes of coordinates is proved and asymptotic approximation of any order of the small parameter is constructed.