Bifurcation diagrams of the Kowalevski top in two constant fields
  The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in R^3. A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels. 
Mikhail P. Kharlamov