  The presence/absence of a Berry phase depends on the topology of the manifold
of dynamical Jahn-Teller potential minima. We describe in detail the relation
between these topological properties and the way the lowest two adiabatic
potential surfaces get locally degenerate. We illustrate our arguments through
spherical generalizations of the linear T x h and H x h cases, relevant for the
physics of fullerene ions. Our analysis allows us to classify all the spherical
Jahn-Teller systems with respect to the Berry phase. Its absence can, but does
not necessarily, lead to a nondegenerate ground state.
