The Lagrange bitop on so(4) x so(4) and geometry of the Prym varieties

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 varietiesof double coverings of hypeelliptic curves. The correspondence between all such coverings with Prym varieties splitted as a sum of two varieties of the same dimension and the integrable hierarchy associated to the initial system is established.