2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/180073We 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 isomonodromy 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 $ τ$-functions or equivalently Toeplitz determinants of the system.to appear J. Approx. TheoryClassical Analysis and ODEsMathematical Physics05E35; 33C45; 34M55; 37K35; 39A05; 42A52text