  Consider the Laguerre polynomials and deform them by the introduction in the
measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx.
Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this
deformation can be described by systems of differential/difference equations
for the corresponding recursion coefficients and that these equations,
ultimately, are equivalent to the Painlev\'e III equation and its
B\"acklund/Schlesinger transformations. Here we prove that an analogue result
holds for some kind of semiclassical matrix-valued orthogonal polynomials of
Laguerre type.
