  The geodesic motion on anti-de Sitter spacetimes is studied pointing out how
the trajectories are determined by the ten independent conserved quantities
associated to the specific SO(2,3) isometries of these manifolds. The new
result is that there are two conserved SO(3) vectors which play the same role
as the Runge-Lenz vector of the Kepler problem, determining the major and minor
semiaxes of the ellipsoidal anti-de Sitter geodesics.
