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A quasilinear autonomous system with an operator of differentiation with respect to the characteristic directions of time and space variables associated with a Lyapunov’s vector field is considered. The question of the existence of multi-periodic solutions on time variables is investigated, when the matrix of a linear system along characteristics has the property of exponential stability. And the non-linear part of the system is sufficiently smooth. In the note, on the basis of Lyapunov’s method, the necessary properties of the characteristics of the system with the specified differentiation operator were substantiated; theorems on the existence and uniqueness of multi-periodic solutions of linear homogeneous and nonhomogeneous systems were proved; sufficient conditions for the existence of a unique multi-periodic solution of a quasilinear system were established. In the study of a nonlinear system, the method of contraction mapping was used.

On multi-periodic solutions of quasilinear autonomous systems with an operator of differentiation on the Lyapunov’s vector field