  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.
