Net diagrams for the Fomenko invariant in an integrable system with three degrees of freedom

The classical Kowalevski problem in the rigid body dynamics is a rare example of a system being generalized to an integrable family of irreducible systems with three degrees of freedom. This generalization is the case of A. G. Reyman and M. A. Semenov-Tian-Shansky (1987). M. P. Kharlamov (2004–2005) has built the stratification of the 6-dimensional phase space by the momentum mapping rank based on detecting all critical subsystems. The problem of description of the Fomenko invariant on iso-energetic surfaces of the Kowalevski top in the double force field was then formulated. The general definition of a topological invariant for integrable systems with arbitrary number of degrees of freedom was introduced in the works of A. T. Fomenko (1988–1991).
In the lecture we illustrate possible approaches to describe Fomenko invariants by means of net diagrams on iso-energetic levels. We start from the classical Kowalevski case and pass to the generalization with the double force field. It shows the effective work of the classification method based on the critical subsystems analysis. For the Kowalevski top in the double force field we present the complete list of the values of the topological invariant in the form of 19 net diagrams on the 5-dimensional iso-energetic levels.