Abstract
Abstract The study is devoted to the application of the small parameter method to the study of the existence of multiperiodic solutions of systems containing small parameter equations with a linear homogeneous partial differential operator with unit coefficients. The operator of the system is called the diagonal differentiation operator of the Euclidean space of independent variables. Such systems, although they have an independent interest in theory in various applications of partial differential equations, they are also a strong tool for studying conditionally periodic solutions of systems of ordinary differential equations. This aspect of the system under consideration is of particular interest to us. Therefore, these systems, both in our considered and in the theory of multi-frequency oscillations, are important as a method for studying conditionally periodic systems of ordinary differential equations. Therefore, such an approach can be called a method for studying conditionally periodic systems of ordinary differential equations with a transition to the study of multiperiodic systems of partial differential equations.