We consider a system of nth-order ordinary differential equations whose right-hand side is the sum of a linear function of the solution with a constant matrix, an essential nonlinearity of the relay type with hysteresis, and a perturbing continuous periodic function. The matrix of the linear function has only real simple nonzero eigenvalues, of which at least one is positive. We study the question of whether such systems have continuous solutions with two switching points in the state space (two-point oscillatory solutions) such that the time in which the solution returns to each of these points coincides with the period of the perturbing function or is an integer fraction of the latter. A sufficient condition for the nonexistence of such solutions is established, and a theorem is proved that gives sufficient conditions for the existence of a two-point oscillatory solution with return time equal to the period of the perturbing function. A corroborating example is given.