The aim of this paper is to prove the existence of a nontrivial solution of the following system of ordinary differential equations: (1) \(\ddot x+V''(x,t)=0\), where \(V=V(x,t)\in C^2(\mathbb{R}^N \times\mathbb{R}, \mathbb{R})\) is a \(T_0\)-periodic function with respect to the variable \(t\), provided that the function \(V_1(x,t): =V(x,t)-{1\over 2} (A_0(t)x,x)\), \(A_0(t)= V''(0,t)\) satisfies some suitable condition at zero. Existence of a nontrivial solution (different from zero) of (1) is based on Morse theory.