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A new method to construct algebro-geometric solutions of rank two Schlesingersystems is presented. For an elliptic curve represented as a ramified doublecovering of CP^1, a meromorphic differential is constructed with the followingproperty: the common projection of its two zeros on the base of the covering,regarded as a function of the only moving branch point of the covering, is asolution of a Painleve VI equation. This differential provides an invariantformulation of a classical Okamoto transformation for the Painleve VIequations. A generalization of this differential to hyperelliptic curves isalso constructed. In this case, positions of zeros of the differential providepart of a solution of the multidimensional Garnier system. The correspondingsolutions of the rank two Schlesinger systems associated with elliptic andhyperelliptic curves are constructed in terms of this differential. The initialdata for construction of the meromorphic differential include a point in theJacobian of the curve, under the assumption that this point has nonvariablecoordinates with respect to the lattice of the Jacobian while the branch pointsvary. It appears that the cases where the coordinates of the point are rationalcorrespond to periodic trajectories of the billiard ordered games associatedwith g confocal quadrics in (g+1)-dimensional space. This is a generalizationof a situation studied by Hitchin, who related algebraic solutions of aPainleve VI equation with the Poncelet polygons.

Algebro-geometric solutions of the Schlesinger systems and the Poncelet-type polygons in higher dimensions