Due to the ubiquitous presence of a Berry phase, in most cases of dynamical Jahn-Teller systems the symmetry of the vibronic ground state is the same as that of the original degenerate electronic state; As a single exception, the linear H x h icosahedral model is determined by an additional free parameter, which can be continuously tuned to eliminate the Berry phase from the low-energy closed paths: accordingly, the ground state changes to a totally symmetric nondegenerate state. The detailed values of the parameters not being known, the issue of the actual ground-state symmetry of C-60(+) is open.