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