Different linearizations of non-abelian second Painlevé systems and related monodromy surfaces

Abstract

In this paper, we discuss a connection between different linearizations for non-abelian analogs of the second Painlevé equation. For each of the analogs, we listed the pairs of the Harnard–Tracy–Widom (HTW), Flaschka–Newell (FN), and Jimbo–Miwa (JM) types. A method for establishing the HTW-JM correspondence is suggested. For one of the non-abelian analogs, we derive the corresponding non-abelian generalizations of the monodromy surfaces related to the FN- and JM-type linearizations. A natural Poisson structure associated with these monodromy surfaces is also discussed.