Parabolicity of degenerate singularities in axisymmetric Euler systems with a gyrostat

We study degenerate singularities of the well-known multiparametric family of integrable Zhukovsky cases of rigid body dynamics, i.e., Euler tops with added constant gyrostatic moment. For an axisymmetric rigid body and systems close to it, it is proved that degenerate local and semilocal singularities are parabolic and cuspidal singularities, respectively, for all values of the set of system parameters, excluding some hypersurfaces. It was checked that these singularities belonging to the preimage of the cusp of the bifurcation curve satisfy the parabolicity criterion of A. V. Bolsinov, L. Guglielmi, and E. A. Kudryavtseva. Therefore, they are structurally stable for small perturbations of the system in the class of integrable systems, in particular, for a small change in the principal moments of inertia, the components of the gyrostatic moment, and the values of the area integral.