Integrability of the generalised Hill problem

We consider a certain two-parameter generalisation of the planar Hill lunar problem. We prove that for nonzero values of these parameters the system is not integrable in the Liouville sense. For special choices of parameters the system coincides with the classical Hill system, the integrable synodical Kepler problem or the integrable parametric Henon system. We prove that the synodical Kepler problem is not super-integrable, and that the parametric Henon problem is super-integrable for infinitely many values of the parameter.