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