The authors study an interesting problem involving periodic solutions having a symmetry induced by an orthogonal matrix \(A\) for which there is an integer \(p\) such that \(A^p\) is the identity. More precisely, given \(T>0\), they look for solutions \(x\) satisfying the relation \(x(t+T)=A x(t)\), for every \(t\). These solutions are \(pT\)-periodic. Some existence results are proposed for first and second order systems, both in the non-resonant and in the resonant cases. However, it seems to me that some assumptions should be added in Section 2 in order to prevent the nonlinearity \(f\) to be, e.g., identically zero.