Cusps without chaos

We present cuspy, non–axisymmetric, scale–free mass models of discs, whose gravitational potentials are of Stäckel form in parabolic coordinates. A black hole may be added at the centre, without in any way affecting the Stäckel form; the dynamics in these potentials is, of course, fully integrable. The surface density, Σdisc ∝ 1/r γ , where 0 < γ < 1 corresponds to steep cusps for which the central force diverges. Thus cusps, black holes, and non–axisymmetry are not a sure recipe for chaos, as is generally assumed. A new family of orbits, lens orbits, emerges to replace the box orbits of models of elliptical galaxies that have constant–density cores. Loop orbits are conspicuous by their absence. Both lenses and boxlets (the other family of orbits), can be elongated in the direction of the density distribution, a property that is favourable for the construction of non–axisymmetric, self–consistent equilibrium models of elliptical galaxies.