Topological atlas of the Kovalevskaya top in a double field

The article contains the rough topological analysis of the completely integrable system with three degrees of freedom corresponding to the motion of the Kovalevskaya top in a double field. This system is not reducible to a family of systems with two degrees of freedom. We introduce the notion of a topological atlas of an irreducible system. For the Kovalevskaya top in a double field, we complete the topological analysis of all critical subsystems with two degrees of freedom and calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the initial momentum map pointing out all chambers, families of $3$-tori, and $4$-atoms of their bifurcations. Basing on the ideas of A. T. Fomenko, we define the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all nondegenerate rank $1$ singularities.