  We study the observational properties of a class of exact solutions of
Einstein's field equations describing stationary, axially symmetric, rigidly
rotating dust (i.e. non interacting particles). We ask the question whether
such solutions can describe astrophysical rotating dark matter clouds near the
center of galaxies and we probe the possibility that they may constitute an
alternative to supermassive black holes at the center of galaxies. We show that
light emission from accretion disks made of ordinary baryonic matter in this
space-time has several differences with respect to the emission of light from
similar accretion disks around black holes. The shape of the iron K$\alpha$
line in the reflection spectrum of accretion disks can potentially distinguish
this class of solution from the Kerr metric, but this may not be possible with
current X-ray missions.
