  We fulfill the rough topological analysis of the problem of the motion of the
Kovalevskaya top in a double field. This problem is described by a completely
integrable system with three degrees of freedom not reducible to a family of
systems with two degrees of freedom. The notion of a topological atlas of an
irreducible system is introduced. The complete topological analysis of the
critical subsystems with two degrees of freedom is given. We calculate the
types of all critical points. We present the parametric classification of the
equipped iso-energy diagrams of the complete 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 introduce the notion of 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 rank 1 singularities. The work was
supported by the grants of the RFBR No. 13-01-97025 and 14-01-00119.
