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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, $Σ_{disc}\propto 1/r^γ$, 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.