  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.
