We consider an autonomous Hamiltonian system $\ddot {q}+V_q(q)=0$ in ${\bf R}^2$, where the potential $V$ has a global maximum at the origin and singularities at some points $\xi_1$, $\xi_2 \in {\bf R}^2 \setminus \{0\}$. Under some compactness conditions on $V$ at infinity and assuming a strong force type condition at the singularities, we study, using variational arguments, the existence of various types of heteroclinic and homoclinic solutions of the system.