  We construct one-parameter families of solutions to the
Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that
the underlying family of spacetimes are each an asymptotically flat,
stationary, axisymmetric, black hole spacetime, and such that the corresponding
scalar fields are non-zero and time-periodic. An immediate corollary is that
for these Klein--Gordon masses, the Kerr family is not asymptotically stable as
a solution to the Einstein--Klein--Gordon equations.
