On canonical parametrization of the phase spaces of equations of isomonodromic deformations of Fuchsian systems of dimension 2x2. Derivation of the Painleve VI equation

This survey considers the factorization, by linear changes of the sought vector-function, of the manifold of 2x2 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-Painleve VI equations, as well as algebro-geometric constructions related to them: the Okamoto surface and a rational atlas of the Darboux coordinates on it.