  We consider the solid-solid interactions in the two body problem. The
relative equilibria have been previously studied analytically and general
motions were numerically analyzed using some expansion of the gravitational
potential up to the second order, but only when there are no direct
interactions between the orientation of the bodies. Here we expand the
potential up to the fourth order and we show that the secular problem obtained
after averaging over fast angles, as for the precession model of Boue and
Laskar [Boue, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but
not trivially. We describe the general features of the motions and we provide
explicit analytical approximations for the solutions. We demonstrate that the
general solution of the secular system can be decomposed as a uniform
precession around the total angular momentum and a periodic symmetric orbit in
the precessing frame. More generally, we show that for a general n-body system
of rigid bodies in gravitational interaction, the regular quasiperiodic
solutions can be decomposed into a uniform precession around the total angular
momentum, and a quasiperiodic motion with one frequency less in the precessing
frame.
