  This paper deals with the classical trajectories for two super-integrable
systems: a system known in quantum chemistry as the Hartmann system and a
system of potential use in quantum chemistry and nuclear physics. Both systems
correspond to ring-shaped potentials. They admit two maximally super-integrable
systems as limiting cases, viz, the isotropic harmonic oscillator system and
the Coulomb-Kepler system in three dimensions. The planarity of the
trajectories is studied in a systematic way. In general, the trajectories are
quasi-periodic rather than periodic. A constraint condition allows to pass from
quasi-periodic motions to periodic ones. When written in a quantum mechanical
context, this constraint condition leads to new accidental degeneracies for the
two systems studied.
