On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables

In the general theory of first order partial differential equations one of the effective ways of integration is a common method of finding solutions by the complete integral [4–5]. In this note we propose a method of research of problems of multiperiodic solutions of linear systems of equations with the same differentiation operator in the direction of a vector field, which is based on the results of the study multiperiodicity of their complete integral. Such an approach is not found in earlier studies based on the methods of fundamental works [8–9] on multiperiodic solutions of such systems. The elements of the proposed method are used in [1–3, 6–7]. In this note in order to establish the multiperiodicity of a complete integral Green's function is introduced in the absence of non-trivial integral multiperiodic solutions of homogeneous system.