  We investigate the electronic transport through a molecule in the Kondo
regime. The tunneling between the electrode and the molecule is asymmetrically
modulated by the oscillations of the molecule, i.e., if the molecule gets
closer to one of the electrodes the tunneling to that electrode will increase
while for the other electrode it will decrease. The system is described by a
two-channel Anderson model with phonon-assisted hybridization, which is solved
with the Wilson numerical renormalization group method. The results for several
functional forms of tunneling modulation are presented. For a linearized
modulation the Kondo screening of the molecular spin is caused by the even or
odd conduction channel. At the critical value of the electron-phonon coupling
an unstable two-channel Kondo fixed point is found. For a realistic modulation
the spin at the molecular orbital is Kondo screened by the even conduction
channel even in the regime of strong coupling. A universal consequence of the
electron-phonon coupling is the softening of the phonon mode and the related
instability to perturbations that break the left-right symmetry. When the
frequency of oscillations decreases below the magnitude of such perturbation,
the molecule is abruptly attracted to one of the electrodes. In this regime,
the Kondo temperature is enhanced and, simultaneously, the conductance through
the molecule is suppressed.
