## On the linearization of the first and second Painlevé equations

### N. Joshi, A. V. Kitaev and P. A. Treharne

#### Abstract

We find Fuchs–Garnier pairs in the form of *3×3*
matrices for the first and second Painlevé equations
which are linear in the spectral parameter. As an application of
our pairs for the second Painlevé equation, we use the
generalized Laplace transform to derive an invertible integral
transformation relating two Fuchs–Garnier pairs in
*2×2* matrices with different singularity structures,
namely, the pair due to Jimbo and Miwa and the one found by
Harnad, Tracy, and Widom. Together with certain other
transformations, it allows us to relate all known
*2×2* matrix Fuchs–Garnier pairs for the second
Painlevé equation with the original Garnier pair.

Keywords:
Painlevé Equations, Isomonodromy Deformations, Laplace Transform, Lax Pair, Stokes Phenomenon.

AMS Subject Classification:
Primary 33E17; secondary 34M25, 34M55.