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 , \Sigma _ { disc } \propto 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 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 .