On the linearization of the first and second Painleve equations

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