Incompatibility of BCS pairing and the Peierls distortion in linear chain systems

We have solved the coupled gap equations for a one dimensional electron gas which can undergo both a Peierls insulating and a BCS superconducting transition. We find that, except for a very special case, (i) only one of the order parameters can be non-vanishing and (ii) there is only one critical temperature. We conclude that, independent of the pairing interaction strength, if the Peierls transition occurs, there can be no BCS superconductivity, and conversely.