Advances in Mathematical Physics
Volume 2009 (2009), Article ID 461860, 14 pages
Research Article

Generalization of Okamoto's Equation to Arbitrary 2×2 Schlesinger System

1Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke West, Montreal QC, H4B 1R6, Canada
2Laboratoire de Physique, Ecole Normale Supérieure de Lyon, Université de Lyon, 46, Allée d'Italie, 69364 Lyon Cedex 07, France

Received 10 June 2009; Accepted 11 September 2009

Academic Editor: Alexander P. Veselov

Copyright © 2009 Dmitry Korotkin and Henning Samtleben. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The 2×2 Schlesinger system for the case of four regular singularities is equivalent to the Painlevé VI equation. The Painlevé VI equation can in turn be rewritten in the symmetric form of Okamoto's equation; the dependent variable in Okamoto's form of the PVI equation is the (slightly transformed) logarithmic derivative of the Jimbo-Miwa tau-function of the Schlesinger system. The goal of this note is twofold. First, we find a universal formulation of an arbitrary Schlesinger system with regular singularities in terms of appropriately defined Virasoro generators. Second, we find analogues of Okamoto's equation for the case of the 2×2 Schlesinger system with an arbitrary number of poles. A new set of scalar equations for the logarithmic derivatives of the Jimbo-Miwa tau-function is derived in terms of generators of the Virasoro algebra; these generators are expressed in terms of derivatives with respect to singularities of the Schlesinger system.