Regular flat structure and generalized Okubo system

Hiroshi Kawakami, Toshiyuki Mano

Published 2017-02-10Version 1

We study a relationship between regular flat structures and generalized Okubo systems. A main result in this paper is that isomonodromic deformations of generically regular generalized Okubo systems can be equipped with flat structures. As an application, we can define flat structures on the spaces of independent variables of (classical) Painlev\'e equations (except for PI). As a bi-product, we can naturally understand the well-known coalescence cascade of the Painlev\'e equations as the degeneration scheme of the Jordan normal forms of a square matrix of rank four.