Topological analysis of axisymmetric Zhukovsky system for the case of the Lie algebra $e(2,1)$

We study an axisymmetric analog of the Zhukovsky integrable case for the Lie algebra $e(2,1)$. Bifurcation diagrams are constructed. They essentially depend both on the constant parameters of the system and on the values of the Casimir functions, which are analogues of the geometric integral and the area integral. The critical set of the system is studied, and the nondegeneracy of its points is checked. Analogues of the Fomenko 3-atoms of the system are determined and it is shown that all of them have the type of direct product of the 2-dimensional base and the 1-dimensional fiber. Non-compact non-critical bifurcations are discovered in the system.