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