First Appelrot class of pseudo-Euclidean Kovalevskaya system

In paper, properties of an integrable pseudo-Euclidean analogue of the Kovalevskaya top are studied for the zero level of the additional first Kovalevskaya integral. The class of motions of a classical top under the same condition is also called the first Appelrot class or the Delaunay class. We describe the homeomorphism class of each fiber, the fiberwise homeomorphism classes of the foliation in a neighborhood of each bifurcation fiber (i.e. analogues of Fomenko 2-atoms) and on the two-dimensional intersection of the level $K = 0$ and each nondegenerate symplectic leaf of the Poisson bracket. It is proved that non-compact one-dimensional Liouville fibers, non-critical bifurcations of compact and non-compact fibers appear in this integrable system. The non-degeneracy problem (in the Bott sense) for all points of the $K = 0$ level is also studied, and it is proved that the critical sets of the of classical Kovalevskaya top and its pseudo-Euclidean analogue coincides.