  We extend similarity reductions of the coupled (2+1)-dimensional three-wave
resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix
Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with
the previously known Fuchs--Garnier pair for the fourth and sixth Painleve'
equations. These Fuchs--Garnier pairs have an important feature: they are
linear with respect to the spectral parameter. Therefore we can apply the
Laplace transform to study these pairs. In this way we found reductions of all
pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and
T. Miwa. As an application of the 3x3 matrix pairs, we found an integral
auto-transformation for the standard Fuchs--Garnier pair for the fifth
Painleve' equation. It generates an Okamoto-like B\"acklund transformation for
the fifth Painleve' equation. Another application is an integral transformation
relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve'
equation.
