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This article concerns the Hamiltonian elliptic system $$\displaylines{ -\Delta u +V(x)u=H_{v}(x, u, v),\quad x\in \mathbb{R}^N, \cr -\Delta v +V(x)v=H_{u}(x, u, v),\quad x\in \mathbb{R}^N, \cr u(x)\to 0,\quad v(x)\to 0, \quad \text{as } |x|\to \infty, }$$ where $z=(u,v): \mathbb{R}^{N}\to\mathbb{R}\times\mathbb{R}$, $N\geq 3$ and the potential V(x) is allowed to be sign-changing. Under weak superquadratic assumptions for the nonlinearities, by applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we obtain the existence of nontrivial and ground state solutions.