  We prove that generic solutions of the vacuum constraint Einstein equations
do not possess any global or local space-time Killing vectors, on an
asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean
curvature close to a constant, or for CMC asymptotically hyperbolic initial
data sets. More generally, we show that non-existence of global symmetries
implies, generically, non-existence of local ones. As part of the argument, we
prove that generic metrics do not possess any local or global conformal Killing
vectors.
