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We derive the Christoffel-Geronimus-Uvarov transformations of a system ofbi-orthogonal polynomials and associated functions on the unit circle, that isto say the modification of the system corresponding to a rational modificationof the weight function. In the specialisation of the weight function to theregular semi-classical case with an arbitrary number of regular singularities $\{z_1, ..., z_M \} $ the bi-orthogonal system is known to be isomonodromypreserving with respect to deformations of the singular points. If the zerosand poles of the Christoffel-Geronimus-Uvarov factors coincide with thesingularities then we have the Schlesinger transformations of thisisomonodromic system. Compatibility of the Schlesinger transformations with theother structures of the system - the recurrence relations, the spectralderivatives and deformation derivatives is explicitly deduced. Various forms ofHirota-Miwa equations are derived for the $ \tau $-functions or equivalentlyToeplitz determinants of the system.

Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II