Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebro-geometric solutions of the Schlesinger systems and the
  Poncelet-type polygons in higher dimensions   A new method to construct algebro-geometric solutions of rank two Schlesinger
systems is presented. For an elliptic curve represented as a ramified double
covering of CP^1, a meromorphic differential is constructed with the following
property: 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 a
solution of a Painleve VI equation. This differential provides an invariant
formulation of a classical Okamoto transformation for the Painleve VI
equations. A generalization of this differential to hyperelliptic curves is
also constructed. In this case, positions of zeros of the differential provide
part of a solution of the multidimensional Garnier system. The corresponding
solutions of the rank two Schlesinger systems associated with elliptic and
hyperelliptic curves are constructed in terms of this differential. The initial
data for construction of the meromorphic differential include a point in the
Jacobian of the curve, under the assumption that this point has nonvariable
coordinates with respect to the lattice of the Jacobian while the branch points
vary. It appears that the cases where the coordinates of the point are rational
correspond to periodic trajectories of the billiard ordered games associated
with g confocal quadrics in (g+1)-dimensional space. This is a generalization
of a situation studied by Hitchin, who related algebraic solutions of a
Painleve VI equation with the Poncelet polygons.

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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
3
The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn