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