Topological Analysis of Axisymmetric Zhukovsky System in the Case of Lie Algebra $$\boldsymbol{e(2,1)}$$

An axisymmetric analog of the Zhukovsky integrable case for the Lie algebra $$e(2,1)$$ is studied. 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 investigated, 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. Noncompact noncritical bifurcations are discovered in the system.