  In this paper we suggest the existence in the central regions of spiral
galaxies of collisionless, scale-free, rigidly rotating, self-gravitating discs
with spiral symmetry. Such discs must be truncated at a finite radius, and they
must be stabilized and rendered self-similar by a suitable halo. The halo and
the rotating disc share the self-similar class and must form together to arrive
at the suggested state. We make comparisons with the well-known rigidly
rotating, Kalnajs discs; one of which is axi-symmetric and finite while the
other is infinite and decomposed into spiral modes. We find the
self-consistent, self-similar, distribution functions in one and two dimensions
in a rigidly rotating, collisionless system. In the case of two dimensions we
deduce the self-consistency condition for discrete spiral arms.
  We give an estimate of the disturbance created in the halo by the presence of
the disc, and argue that the halo itself should be close to self-similarity. A
very weak cusp in the halo may be necessary. The necessary spatial coincidence
of the halo results in a kind of disc-halo `conspiracy'. Finally the disc
equations are formulated in `spiral' coordinates, and the passage to an
approximately discrete `line spiral' is given as an example. Although in two
dimensions the collisionless particles enter and leave the arms in non-linear
epicycles, they move approximately parallel to the arms in the line spiral
limit. The spiral pattern is however in rigid rotation. Aperiodic spiral arms
are suggested wherein discontinuities may be coarse-grained to appear as
collisionless shocks.
