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: Mumford's relation and Mumford-Dalalyan theory. 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.