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 {z1,…,zM}{z1,…,zM} 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 ττ-functions or equivalently Toeplitz determinants of the system.