Can galactic nuclei be non-axisymmetric? --- The parameter space of power-law discs

The shape of a cusped galactic nucleus is constrained by the range of shapes of orbits in its gravitational potential. It is shown for scale-free non-axisymmetric discs that while a plausible elongated density model requires at least some orbits to spend more time near the major axis than anywhere else, both regular boxlets and tube orbits generally cross the major axis too fast for self-consistency. If galaxies host inner nuclear discs or flat bars with a cuspy surface light profile ($\gamma=|{d \log \mu / d \log r}|>0$), their ellipticity $1-{b / a}$ cannot be greater than about ${\gamma / 2}$. Discs or bars with a shallow central profile ($\gamma \le 0.3$) should not be strongly elliptical.