  Aims. In this paper we continue our study of density cusps that may contain
central black holes. Methods. We recall our attempts to use distribution
functions with a memory of self-similar relaxation, but mostly they apply only
in restricted regions of the global system. We are forced to consider related
distribution functions that are steady but not self-similar. Results. One
remarkably simple distribution function that has a filled loss cone describes a
bulge that transits from a near black hole domain to an outer 'zero flux'
regime where$\rho\propto r^{-7/4}$. The transition passes from an initial
inverse square profile through a region having a 1/r density profile. The
structure is likely to be developed at an early stage in the growth of a
galaxy. A central black hole is shown to grow exponentially in this background
with an e-folding time of a few million years. Conclusions. We derive our
results from first principles, using only the angular momentum integral in
spherical symmetry. The initial relaxation probably requires bar instabilities
and clump-clump interactions.
