Critical solution for a Hillʹs type problem

We study the problem of two satellites attracted by a center of force. Assuming that the motion of the center of mass of the two satellites is a Keplerian circular orbit around the center of force, we regularize the collision between them using the Levi-Civita procedure. The existence of a constant of motion in the extended phase space allows us to study the stability of the solution, where the two satellites are tied together in their circular motion around the center of force. We call this solution the critical solution. A theorem of M. Kummer is applied to prove, in specific conditions, the existence of two one-parametric families of almost periodic orbits for the satellites motion that bifurcates from the critical solution.