  We study the Josephson current through a resonant level coupled to a
vibration mode (local Holstein model) in the adiabatic limit of low oscillator
frequency. A semiclassical theory is then appropriate and allows us to consider
the oscillator dynamics within the Born-Oppenheimer approximation for arbitrary
electron-vibration couplings. The resulting Fokker-Planck equation has been
solved in the most relevant underdamped limit and yields the oscillator
distribution function and the Josephson current. Remarkably, a transition from
single-well to double-well behavior of the effective oscillator potential
surface is possible and can be tuned by variation of the superconducting phase
difference. The Josephson current is shown to be only weakly affected by the
electron-vibration coupling due to strong phonon localization near the bottom
of the potential surface.
