  We show through selected examples, relevant to the physics of fullerene ions,
that the presence of a Berry phase in dynamical Jahn-Teller systems does not
guarantee the degeneracy of the ground state, contrary to what previously
believed. Moreover, we introduce a new class of Jahn-Teller systems with no
Berry phase, as a generalization of the basic icosahedral $H \otimes h$ case.
