  Due to the frequent presence of a Berry phase, in most cases of dynamical
Jahn-Teller systems the symmetry of the ground state is the same as that of the
electronic state. However, the H x h icosahedral case, relevant for the physics
of fullerene ions, provides a first example of linear coupling leading, at
strong coupling, to a change in symmetry of the ground state to a totally
symmetric nondegenerate state. We generalize this observation and show through
detailed examples that the absence of a Berry phase can, but does not
necessarily, lead to a nondegenerate ground state.
