Communications in Mathematical Physics

, Volume 369, Issue 2, pp 403–431 | Cite as

Regular Flat Structure and Generalized Okubo System

  • Hiroshi Kawakami
  • Toshiyuki ManoEmail author


We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlevé equations (except for PI). In our framework, the Painlevé equations PVI–PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlevé equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four.


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The authors thank Professor C. Hertling for reading a preprint and giving useful comments on it and Professor M. Noumi for useful discussions. This work was partially supported by JSPS KAKENHI Grant Nos. 25800082, 17K05335.


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Authors and Affiliations

  1. 1.Department of Physics and Mathematics, College of Science and EngineeringAoyama Gakuin UniversitySagamihara-shiJapan
  2. 2.Department of Mathematical Sciences, Faculty of ScienceUniversity of the RyukyusNishihara-choJapan

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