  We prove that if a stationary, real analytic, asymptotically flat vacuum
black hole spacetime of dimension $n\geq 4$ contains a non-degenerate horizon
with compact cross sections that are transverse to the stationarity generating
Killing vector field then, for each connected component of the black hole's
horizon, there is a Killing field which is tangent to the generators of the
horizon. For the case of rotating black holes, the stationarity generating
Killing field is not tangent to the horizon generators and therefore the
isometry group of the spacetime is at least two dimensional. Our proof relies
on significant extensions of our earlier work on the symmetries of spacetimes
containing a compact Cauchy horizon, allowing now for non closed generators of
the horizon.
