In this paper, we study the problem of finding solutions to a system of second-order partial differential equations related to hypergeometric functions of four variables F₁₆, F₁₈, F_19, F₂₀ and F₃₁. Currently, mathematical modeling of physical processes and forecasting their dynamics are becoming increasingly important in science and technology. Most of these processes are described by differential equations or their systems, but only a few of them, those that reflect real physical phenomena, allow analytical solutions expressed in terms of elementary functions. This fact makes it particularly relevant to study more general classes of functions, in particular hypergeometric functions of many variables. Hypergeometric functions are solutions of certain systems of differential equations and belong to the category of special or transcendental functions. Due to their properties, they are widely used in solving multidimensional problems, and the study of their analytical structure is important both for the development of theoretical mathematics and for practical calculations. Within the framework of this study, linearly independent solutions of these systems were constructed for the functions under consideration, as well as their structural features and main characteristics were analyzed. The results obtained form the theoretical basis for further study of hypergeometric functions of a higher number of variables and open up prospects for their effective use in solving applied problems. The work contributes to the development of the theory of special functions and offers new approaches to solving fundamental problems of mathematical physics and applied mathematics.