We study an n-dimensional system of ordinary differential equations with hysteresis type relay nonlinearity and a periodic perturbation function on the right-hand side. It is supposed that the matrix of the system is real and symmetric and, moreover, it has an eigenvalue of multiplicity two. In the phase space of the system, we consider continuous bounded oscillatory solutions with two fixed points and the same time of return to each of these points. For these solutions, we prove the existence and nonexistence theorems. For a three-dimensional system, these results are illustrated by a numerical example.