On Hamiltonian potentials with quartic polynomial normal variational equations

In this paper we prove that there exists only one family of classical Hamiltonian systems of two degrees of freedom with invariant plane $\mathrm{\Gamma }=\{q_2=p_2=0\}$ whose normal variational equation around integral curves in $\mathrm{\Gamma }$ 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.