  In this paper we prove that there exists only one family of classical
Hamiltonian systems of two degrees of freedom with invariant plane
$\Gamma=\{q_2=p_2=0\}$ whose normal variational equation around integral curves
in $\Gamma$ is generically a Hill-Schr\"odinger equation with quartic
polynomial potential. In particular, by means of the Morales-Ramis theory,
these Hamiltonian systems are non-integrable through rational first integrals.
