We study a system of ordinary differential equations of order n containing a nonlinearity of imperfectrelay type with hysteresis and external periodic perturbation. We consider the problem of existence of solutions with periods equal to (or multiple of) the period of external perturbation with two points of switching within the period. The problem is solved in the case where the collection of simple real eigenvalues of the matrix of the system contains an eigenvalue equal to zero. By a nonsingular transformation, the system is reduced to a canonical system of special form, which enables us to perform its analysis by the analytic methods. We propose an approach to finding the points of switching for the representative point of periodic solution and to the choice of the parameters of nonlinearity and the feedback vector. A theorem on necessary conditions for the existence of periodic solutions of the system is proved. Sufficient conditions for the existence of the required solutions are established. We also perform the analysis of stability of solutions by using the point mapping and the fixed-point method. We present an example that confirms the established results.