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 counterargument to this conclusion. We consider 2D analogues of triaxial cusps: a sequence of non-axisymmetric, cuspy discs first described by Sridhar & Touma. Scale-free models with potential Phi proportional to r(alpha) are examined in detail. It is shown analytically for 0 < alpha less than or similar to 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. Together with existing work on cusps, we conclude on purely theoretical grounds that galactic nuclei are not expected to be triaxial.