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We present an algebraic geometrical and analytical description of theGoryachev case of rigid body motion. It belongs to a family of systems sharingthe same properties: although completely integrable, they are not algebraicallyintegrable, their solution is not meromorphic in the complex time and involvesdynamics on the strata of the Jacobian varieties of trigonal curves.  Although the strata of hyperelliptic Jacobians have already appeared in theliterature in the context of some dynamical systems, the Goryachev case is thefirst example of an integrable system whose solution involves a more generalcurve. Several new features (and formulae) are encountered in the solutiongiven in terms of sigma-functions of such a curve.

Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion