The number of connected components in the preimage of a regular value of the momentum mapping for the geodesic flow on ellipsoid

A Liouville foliation of the geodesic flow of a generic ellipsoid is considered in the paper. The main goal is the demonstration of various approaches to computation of the number of connected components in the preimage of a regular value of the moment map. This is done with the use of the Boolean functions method of M. P. Kharlamov and also N. T. Zung's result on the decomposition of a hyperbolic singularity to an almost direct product of $2$-dimensional atoms.