Density cusps: restrictions on non-axisymmetric models

Galactic nuclei are now generally thought to have density cusps in their centres, and the evidence is mounting that as a consequence they are unlikely to be triaxial. Self-consistent stellar dynamical models of non-axisymmetric cusps would be an interesting counter-argument to this conclusion. We consider 2-d analogues of triaxial cusps: a sequence of non-axisymmetric, cuspy discs first described by Sridhar & Touma (1997). Scale-free models with potential {}$\mathrm{\Phi }\propto r^{\alpha }$ are examined in detail. It is shown analytically for $0<\alpha \text{la}0.43$ that self-consistent models with positive phase-space density do not exist. Numerical solutions of the combined Vlasov and Poisson equations suggest that the whole sequence of models with $0<\alpha <1$ are also unphysical. Along with existing work on cusps, we conclude on purely theoretical grounds that galactic nuclei are not expected to be triaxial.