Generalized Hill's problem: Some cases of complete integrability

In this paper, we are investigating cases of integrability in the planar Hill's problem. The external potential U _extis supposed to be time independent in a given uniformly rotating frame. Cases of integrability of the relative motion of two interacting particles in the vicinity of an equilibrium solution of U _extare found. In all these cases, the form of the second integral is explicitly given, the first being the Jacobian one. Cases in which the interacting potential U between the two particles is of newtonian type are particularized.