Non-Integrability Criteria for Hamiltonians in the Case of Lamé Normal Variational Equations

We consider complex analytic classical Hamiltonian systems with two degrees of freedom and an invariant plane. Furthermore we assume that the normal variational equations (NVE) are of Lamé type. We describe the possible potentials giving rise to this kind of problems and non-integrability criteria based on the differential Galois approach to the Ziglin theorem. Some examples are included.