Three dimensional, axisymmetric cusps without chaos

We construct three dimensional axisymmetric, cuspy density distributions, whose potentials are of St"ackel form in parabolic coordinates. As in Sridhar and Touma (1997), a black hole of arbitrary mass may be added at the centre, without destroying the St"ackel form of the potentials. The construction uses a classic method, originally due to Kuzmin (1956), which is here extended to parabolic coordinates. The models are highly oblate, and the cusps are “weak”, with the density, $\rho \propto 1\mathrm{/}{r}^k$, where $0<k<1$.