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.1 These can be related to other Jacobians in the case of dimension 2.where J = (J 1 , J 2 , J 3 ) T and γ = (γ 1 , γ 2 , γ 3 ) T are the angular and linear momentum respectively and H(J, γ) is the Hamiltonian, which is also a first integral. In addition to the Hamiltonian the equations always possess the integrals (Casimir functions) C 1 = J, γ , C 2 = γ, γ . Apart from the well known integrable cases of Kirchhoff, Clebsch, Steklov and Lyapunov (and their gyroscopic generalizations) there are some further special cases of integrability cases where an additional integral exists only under the condition C 1 = 0. In the most classical case found by D. Goryachev [Gor912] and S. Chaplygin [Chap904] the extra integral is cubic in J. In this case Chaplygin himself [Chap904] gave a separation of variables and reduced the system to quadratures containing integrals on a hyperelliptic genus 2 curve. A detailed algebro-geometric description of the complex invariant manifolds was made in [BvM987]. Further rather exotic special cases of integrability also exist for which neither separation of variables nor explicit solution were known until recently. Here we concentrate on the Goryachev case [Gor916] which was reduced to quadratures in [VT09] by using a bi-Hamiltonian structure and the corresponding separating Darboux-Nijenhuis variables.2 There are several exceptions in the family: one of them is the classical Goryachev-Chaplygin system [Chap904, Gor916], which is linearized on Jacobians of genus 2 hyperelliptic curve. DYNAMICS ON STRATA OF TRIGONAL JACOBIANS 3 The Goryachev case, the focus of this paper, has Hamiltonian H 1 = H and extra integral H 2 that take the form H 1 = J 2 1 + J 2 2 + 4 3 J 2 3 + aγ 1 + b γ 2/3 3 , a, b being arbitrary constants,