  The contributions of Sophya Kowalewski to the integrability theory of the
equations for the heavy top extend to a larger class of Hamiltonian systems on
Lie groups; this paper explains these extensions, and along the way reveals
further geometric significance of her work in the theory of elliptic curves.
Specifically, in this paper we shall be concerned with the solutions of the
following differential system in six variables h_1,h_2,h_3,H_1,H_2,H_3
  dH_1/dt = H_2 H_3 (1/c_3 - 1/c_2) + h_2 a_3 - h_3 a_2,
  dH_2/dt = H_1 H_3 (1/c_1 - 1/c_3) + h_3 a_1 - h_1 a_3,
  dH_3/dt = H_1 H_2 (1/c_2 - 1/c_1) + h_1 a_2 - h_2 a_1,
  dh_1/dt = h_2 H_3/c_3 - h_3 H_2/c_2 + k (H_2 a_3 - H_3 a_2),
  dh_2/dt = h_3 H_1/c_1 - h_1 H_3/c_3 + k (H_3 a_1 - H_1 a_3),
  dh_3/dt = h_1 H_2/c_2 - h_2 H_1/c_1 + k (H_1 a_2 - H_2 a_1),
  in which a_1,a_2,a_3,c_1,c_2,c_3 and k are constants.
