The article considers the issues of existence and uniqueness of the solution in the broad sense of a system of differential equations in partial derivatives of the first order with the same main part with periodic conditions in part of variables. In many mathematical models, especially in the theory of "fine water," the movement of an uncompressible liquid in shallow channels, a flat steady supersonic flow of a compressible gas, differential partial differential equations arise. Classical solutions of nonlinear equations have the property of an unlimited increase in the magnitude of derivatives, which is called a gradient catastrophe (a shock wave formed from a compression wave). The meaning of this property is that with arbitrarily smooth initial values, the first derivative solutions remain limited, only for a finite time. Therefore, there is an urgent need to expand the concept of classical solutions to first-order partial differential equation systems. The article establishes sufficient conditions for the existence and uniqueness of the solution in the broad sense of systems of partial differential equations with the same main part with periodic conditions in terms of variables.