e.g. [@ll], [@fo] for the monopole case, and [@dam], [@ds] for the pole-dipole case). These will provide useful checks on our general results. Another limiting situation of possible relevance is that of a test point-like mass moving in the gravitational field of $N-1$ extended bodies. The motion of the artificial/natural solar system satellites is a typical case of this category. Ref. [@dam4] computed explicitly the form of the corresponding Lagrangian in terms of the gravitational potentials $(W,W_a)$, which can be algorithmically constructed using the formulas given in the Appendix of [@dam2]. However, in this case we have first integrals only when the Lagrangian (describing geodesic motion) possesses some continuous symmetries (Noether’s theorem). For instance, we can consider $N=2$ with $M_2 << M_1$ (restricted two body problem) and with a stationary and/or axially symmetric central body (see e.g. Ref. [@heim] for a study of a restricted problem of this type). This type of limiting situation will not give us useful checks.

Finally, let us remark that we hope that the present work will find practical applications in the relativistic motion of binary stars or in the celestial mechanics of the solar system. Let us recall, for instance, that one simplifies the 