  We study the class of nonholonomic mechanical systems formed by a heavy
symmetric ball that rolls without sliding on a surface of revolution, which is
either at rest or rotates about its (vertical) figure axis with uniform angular
velocity. The first studies of these systems go back over a century, but a
comprehensive understanding of their dynamics is still missing. The system has
an $SO(3) \times SO(2)$ symmetry and reduces to four dimensions. We extend in
various directions, particularly from the case $\Omega = 0$ to the case $\Omega
\ne 0$, a number of previous results and give new results. In particular, we
prove that the reduced system is Hamiltonizable even if $\Omega \ne 0$ and,
exploiting the recently introduced `moving energy', we give sufficient
conditions on the profile of the surface that ensure the periodicity of the
reduced dynamics and hence the quasi-periodicity of the unreduced dynamics on
tori of dimension up to three. Furthermore, we determine all the equilibria of
the reduced system, which are classified in three distinct families, and
determine their stability properties. In addition to this, we give a new form
of the equations of motion of nonholonomic systems in quasi-velocities which,
at variance from the well known Hamel equations, use any set of
quasi-velocities and explicitly contain the reaction forces.
