  The study of self-gravitating stellar systems has provided important hints to
develop tools of analytical mechanics. In the present contribution we review
how to exploit detuned resonant normal forms to extract information on several
aspects of the dynamics in systems with self-similar elliptical equipotentials.
In particular, using energy and ellipticity as parameters, we compute the
instability thresholds of axial orbits, bifurcation values of low-order boxlets
and phase-space fractions pertaining to the families around them. We also show
how to infer something about the singular limit of the potential.
