  The interplay between vibrational modes and Kondo physics is a fundamental
aspect of transport properties of correlated molecular conductors. We present
theoretical results for a single molecule in the Kondo regime connected to left
and right metallic leads, creating the usual coupling to a conduction channel
with left-right parity ("even"). A center-of-mass vibrational mode introduces
an additional, phonon-assisted, tunneling through the antisymmetric ("odd")
channel. A non-Fermi liquid fixed point, reminiscent of the two-channel Kondo
effect, appears at a critical value of the phonon-mediated coupling strength.
Our numerical renormalization-group calculations for this system reveal
non-Fermi-liquid behavior at low temperatures over lines of critical points.
Signatures of this strongly correlated state are prominent in the thermodynamic
properties and in the linear conductance.
