Parabolicity of degenerate singularities of axisymmetric Zhukovsky case

The degenerate singularities of systems from one well-known multiparameter family of integrable systems of rigid body dynamics are studied. Axisymmetric Zhukovsky systems are considered, i.e. axisymmetric Euler tops after adding a constant gyrostatic moment. For all values of the set of parameters, excluding some hypersurfaces, it is proved that the degenerate local and semi-local singularities of the system are of the parabolic and cuspidal type, respectively. Thus these singularities are structurally stable under small perturbations of the system in the class of integrable systems. Note that all these degenerate points lie in the preimage of the cusp point of the parametric bifurcation curve and satisfy the criterion of A.~Bolsinov, L.~Guglielmi and E.~Kudryavtseva.