  Here we apply the general scheme for description of the mechanics of
infinitesimal bodies in the Riemannian spaces to the examples of geodetic and
non-geodetic (for two different model potentials) motions of infinitesimal
rotators on the Mylar balloons. The structure of partial degeneracy is
investigated with the help of the corresponding Hamilton-Jacobi equation and
action-angle analysis. In all situations it was found that for any of the six
disjoint regions in the phase space among the three action variables only two
of them are essential for the description of our models at the level of the old
quantum theory (according to the Bohr-Sommerfeld postulates). Moreover, in both
non-geodetic models the action variables were intertwined with the quantum
number $N$ corresponding to the quantization of the radii $r$ of the inflated
Mylar balloons.
