  A Theorem is proved which reduces the problem of completeness of orbits of
Killing vector fields in maximal globally hyperbolic, say vacuum, space--times
to some properties of the orbits near the Cauchy surface. In particular it is
shown that all Killing orbits are complete in maximal developements of
asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact
manifold. This result gives a significant strengthening of the uniqueness
theorems for black holes.
