Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems - II

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z"1,...,z"M} the bi-orthogonal system is known to be monodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the @t-functions or equivalently Toeplitz determinants of the system.