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Our work studies sequences of orthogonal polynomials $\{P_{n}(x)\}_{n=0}^{\infty} $ of the Laguerre-Hahn class, whose Stieltjesfunctions satisfy a Riccati type differential equation with polynomialcoefficients, are subject to a deformation parameter $t$. We derive systems ofdifferential equations and give Lax pairs, yielding non-linear differentialequations in $t$ for the recurrence relation coefficients and Lax matrices ofthe orthogonal polynomials. A specialisation to a non semi-classical caseobtained via a M\"{o}bius transformation of a Stieltjes function related to amodified Jacobi weight is studied in detail, showing this system is governed bya differential equation of the Painlev\'e type P$_\textrm{VI}$. The particularcase of P$_\textrm{VI}$ arising here has the same four parameters as thesolution found by Magnus [A.P. Magnus, Painlev\'e-type differential equationsfor the recurrence coefficients of semi-classical orthogonal polynomials, J.Comput. Appl. Math., 57:215-237, 1995] but differs in the boundary conditions.

Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials