The authors consider the following first-order Hamiltonian system
where is the standard symplectic matrix in , and has the form with being a symmetric constant matrix, and . The main result of the paper shows, by using two recent critical point theorems for strongly indefinite functionals [T. Bartsch and Y. Ding, Math. Nachr. 279, No. 12, 1267–1288 (2006; Zbl 1117.58007)], that if the technical working assumptions , hold, then the considered Hamiltonian system has at least one homoclinic orbit (Theorem 1.1).