Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations.