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

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 1, pp. 45-127

Résumé

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.

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