A first integral to the partially averaged Newtonian potential of the three-body problem

We consider the partial average i.e., the Lagrange average with respect to {\it just one} of the two mean anomalies, of the Newtonian part of the perturbing function in the three–body problem Hamiltonian. We prove that such a partial average exhibits a non–trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint r^ole (together with certain symmetries) in the appearance of the so called “Herman resonance”. Finally, we discuss an application and an open problem.