  We analyze the interplay between vibrational and electronic degrees of
freedom in charge transport across a molecular single-electron transistor. We
focus on the wide class of molecules which possess quasi-degenerate vibrational
eigenstates, while no degeneracy occurs for their anionic configuration. We
show that the combined effect of a thermal environment and coupling to leads,
involving tunneling events charging and discharging the molecule, leads to a
dynamical symmetry breaking where quasi-degenerate eigenstates acquire
different occupations. This imbalance gives rise to a characteristic asymmetry
of the current versus an applied gate voltage.
