## On the algebro-geometric integration of the Schlesinger equations

A new approach to the construction of isomonodromy deformations of 2 × 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given number 2g + 1, g ≥ 1, of finite (regular) singularities, the method produces a 2g-parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann-Hilbert problem 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 corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g = 1 the solution found coincides with the general elliptic solution of the particular case of the Painlevé VI equation first obtained by N. J. Hitchin [H1].

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- Mathematical Physics
- 0206 Quantum Physics
- 0105 Mathematical Physics
- 0101 Pure Mathematics

### Citation

*Communications in Mathematical Physics*,

*203*(3), 613–633. https://doi.org/10.1007/s002200050037

*Communications in Mathematical Physics*203, no. 3 (January 1, 1999): 613–33. https://doi.org/10.1007/s002200050037.

*Communications in Mathematical Physics*, vol. 203, no. 3, Jan. 1999, pp. 613–33.

*Scopus*, doi:10.1007/s002200050037.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Mathematical Physics
- 0206 Quantum Physics
- 0105 Mathematical Physics
- 0101 Pure Mathematics