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The Lagrange bitop on \(\text{so}(4)\times \text{so}(4)\) and geometry of the Prym varieties. (English) Zbl 1125.37316
Summary: A four-dimensional integrable rigid-body system is considered and it is shown that it represents two twisted three-dimensional Lagrange tops. A polynomial Lax representation, which doesn’t fit neither in Dubrovin’s nor in Adler–van Moerbeke’s picture is presented. The algebro-geometric integration procedure is based on deep facts from the geometry of the Prym varieties of double coverings of hyperelliptic curves. The correspondence between all such coverings with Prym varieties split as a sum of two varieties of the same dimension and the integrable hierarchy associated to the initial system is established.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
14H40 Jacobians, Prym varieties
70E45 Higher-dimensional generalizations in rigid body dynamics
70G55 Algebraic geometry methods for problems in mechanics
70E40 Integrable cases of motion in rigid body dynamics
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