Cusps without chaos

We present cuspy, non-axisymmetric, scale-free mass models of discs, whose gravitational potentials are of Stackel form in parabolic coordinates. A black hole may be added at the centre, without in any way affecting the Stackel form; the dynamics in these potentials is, of course, fully integrable. The surface density Sigma(disc) proportional to 1/r(gamma), where 0<gamma<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 bananas (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.