Here we find the complete family of two degree of freedom classical Hamiltonians with invariant plane $\Gamma=\{q_2=p_2=0\}$ whose normal variational equation around integral curves in $\Gamma$ is a generically a Hill-Schr\"odinger equation with cuartic polynomial potential. In particular, these Hamiltonian form a family of non-integrable Hamiltonians through rational first integrals.

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