In this paper, we consider the following first-order Hamiltonian system $$\dot{z} = \mathcal{J} H_{z}(t,z),$$ where $H\in C^{1}(\mathbb{R}\times\mathbb{R}^{2N},\mathbb{R})$ isthe form $H(t,z)=\frac{1}{2}L(t)z\cdot z + R(t,z)$. By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we establish nontrivial and ground state solutions for the above system under conditions weaker than those in [39].