Consider the second order Hamiltonian system
where , is -periodic and it has the form with a continuous -periodic symmetric matrix, and the standard inner product in . The authors study the existence of infinitely many nontrivial solutions of this system using the variant fountain theorems obtained by W. Zou [Manuscr. Math. 104, No. 3, 343–358 (2001; Zbl 0976.35026)], under the hypothesis that is even in the variable .