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We consider two Lax systems for the homogeneous Painlev\'{e} II equation: oneof size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and oneof size $4\times 4$ introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens inthe early 2010's. We prove that solutions to the $4\times 4$ system can bederived from those to the $2\times 2$ system via an integral transform, andconsequently relate the Stokes multipliers for the two systems. As corollarieswe are able to express two kernels for determinantal processes as contourintegrals involving the Flaschka-Newell Lax system: the tacnode kernel arisingin models of nonintersecting paths, and a critical kernel arising in atwo-matrix model.

Two Lax systems for the Painlev\'e II equation, and two related kernels in random matrix theory