Abstract <p> The paper is devoted to the study of a multidimensional system of ordinary differential equations of the first order with discontinuous nonlinearity of the nonideal relay type and a perturbation in the form of a truncated Fourier series. The continuous oscillatory solutions are studied with two points in the phase space of the system that are chosen on the discontinuity surfaces and an arbitrary period of return to each of these points. The Cauchy problem is solved using the fitting method. A nonsingular transformation is used that reduces the matrix of the system with real simple nonzero eigenvalues to a diagonal form. Estimates are obtained for the solution to the transformed system with a feedback vector of a special form for the case in which the return period is commensurate with the period of the perturbation function but is not a multiple of it. Conditions are established for the parameters of the system and of the solution for which the phase trajectory of the solution is in a bounded domain between the discontinuity surfaces and when the trajectory intersects the surfaces. An example is given to illustrate the theoretical results. </p>