On the algebro-geometric integration of the Schlesinger equations

A new approach to the construction of isomonodromy deformations of 2 x 2 Fu chsian systems is presented. The method is based on a combination of the al gebrogeometric scheme and Riemann-Hilbert approach of the theory of integra ble systems. For a given number 2g + 1, g greater than or equal to 1, of fi nite (regular) singularities, the method produces a 2g-parameter submanifol d of the Fuchsian monodromy data for which the relevant Riemann-Hilbert pro blem can be solved in closed form via the Baker-Akhiezer function technique . This in turn leads to a 2g-parameter family of solutions of the correspon ding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g = 1 the solution found coincides with t he general elliptic solution of the particular case of the Painleve VI equa tion first obtained by N. J. Hitchin [H1].