  In the framework of a simple gravitational theory that contains a scalar
field minimally coupled to gravity, we investigate the emergence of analytic
black-hole solutions with non-trivial scalar hair of secondary type. Although
it is possible for one to obtain asymptotically (A)dS solutions using our
setup, in the context of the present work, we are solely interested in
asymptotically flat solutions. At first, we study the properties of static and
spherically symmetric black-hole solutions emanating from both regular and
phantom scalar fields. We find that the regular-scalar-field-induced solutions
are solutions describing ultra-compact black holes, while the phantom scalar
fields generate ultra-sparse black-hole solutions. The latter are black holes
that can be potentially of very low density since, contrary to ultra-compact
ones, their horizon radius is always greater than the horizon radius of the
corresponding Schwarzschild black hole of the same mass. Then, we generalize
the above static solutions to slowly rotating ones and compute their angular
velocities explicitly. Finally, the study of the axial perturbations of the
derived solutions takes place, in which we show that there is always a region
in the parameter space of the free parameters of our theory that allows the
existence of both ultra-compact and ultra-sparse black holes.
