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We extend similarity reductions of the coupled (2+1)-dimensional three-waveresonant interaction system to its Lax pair. Thus we obtain new 3x3 matrixFuchs--Garnier pairs for the third and fifth Painleve' equations, together withthe previously known Fuchs--Garnier pair for the fourth and sixth Painleve'equations. These Fuchs--Garnier pairs have an important feature: they arelinear with respect to the spectral parameter. Therefore we can apply theLaplace transform to study these pairs. In this way we found reductions of allpairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo andT. Miwa. As an application of the 3x3 matrix pairs, we found an integralauto-transformation for the standard Fuchs--Garnier pair for the fifthPainleve' equation. It generates an Okamoto-like B\"acklund transformation forthe fifth Painleve' equation. Another application is an integral transformationrelating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve'equation.

On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System