We study an n-dimensional system of ordinary differential equations with a hysteresis type relay nonlinearity and a periodic
perturbation function in the right-hand side. It is supposed that the matrix of the system is real and symmetric and 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 return time to each of these points. For such solutions, we prove the existence and
nonexistence theorems. These results are illustrated by a numerical example for a three-dimensional system