  In 1981 Wyman classified the solutions of the Einstein--Klein--Gordon
equations with static spherically symmetric spacetime metric and vanishing
scalar potential. For one of these classes, the scalar field linearly grows
with time. We generalize this symmetry noninheriting solution, perturbatively,
to a rotating one and extend the static solution exactly to arbitrary spacetime
dimensions. Furthermore, we investigate the existence of nonminimally coupled,
time-dependent real scalar fields on top of static black holes, and prove a
no-hair theorem for stealth scalar fields on the Schwarzschild background.
