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We consider an open connected set $\Omega$ and a smooth potential $U$ whichis positive in $\Omega$ and vanishes on $\partial\Omega$. We study theexistence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] thatconnect different components of $\partial\Omega$ and lie on the zero level ofthe energy. We allow that $\partial\Omega$ contains a finite number of criticalpoints of $U$. The case of symmetric potential is also considered.

On the existence of connecting orbits for critical values of the energy