New $$2\times 2$$ 2 × 2 -Matrix Linear Problems for the Painlevé Equations III, V

Abstract

We construct $$2\times 2$$ -matrix linear problems with a spectral parameter for the Painleve equations I–V by means of the degeneration processes from the elliptic linear problem for the Painleve VI equation. These processes supplement the known degeneration relations between the Painleve equations with the degeneration scheme for the associated linear problems. The degeneration relations constructed in this paper are based on the trigonometric, rational, and Inozemtsev limits. The obtained $$2\times 2$$ -matrix linear problems for the Painleve equations III and V are new.