This article is cited in 12 papers

On canonical parametrization of the phase spaces of equations of isomonodromic deformations of Fuchsian systems of dimension $2\times 2$. Derivation of the Painlevé VI equation

M. V. Babich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: This survey considers the factorization, by linear changes of the sought vector-function, of the manifold of $2\times 2$ matrix linear differential equations of first order with simple poles on the right-hand side. It is shown how under a parametrization of such quotient manifolds there naturally appear the Garnier–Painlevé VI equations, as well as algebro-geometric constructions related to them: the Okamoto surface and a rational atlas of the Darboux coordinates on it.

UDC: 517.912

MSC: Primary 34-02, 34M55; Secondary 33E17, 34A26, 34A30, 34M35, 37J05

Received: 20.10.2008

DOI: 10.4213/rm9261

 English version: Russian Mathematical Surveys, 2009, 64:1, 45–127

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