The authors give sufficient conditions for the existence of homoclinic and heteroclinic orbits of Hamiltonian systems \[ u''-L(t)u+V_u(t,u)=0,\tag{*} \] where \(L\) is a symmetric positive definite \(n\times n\) matrix and the potential \(V\) is supposed to be superquadratic in \(u\). The system (*) is first studied on a bounded interval \((-T,T)\) with the boundary conditions \(u(-T)=0=u(T)\) (the existence of a nontrivial solution is proved via the mountain pass lemma) and then the limiting process \(T\to \infty \) is used. A similar approach to an investigation of orbits of (*) is used by \textit{P. Korman} and \textit{A. C. Lazer} [Electron. J. Differ. Equ., 1994/01 (1994; Zbl 0788.34042)] but under the assumption that \(L\) and \(V\) are even functions of \(t\).