  Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show
that two $(n+1)$-dimensional stationary vacuum space-times (possibly with
cosmological constant $\Lambda \in \R$) that coincide up to order one along a
timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal
T$. We further prove that KIDS of $\partial M$ extend to Killing vectors near
$\partial M$. In the AdS type setting, we show unique continuation near
conformal infinity if the metrics have the same conformal infinity and the same
undetermined term. Extension near $\partial M$ of conformal Killing vectors of
conformal infinity which leave the undetermined Fefferman-Graham term invariant
is also established.
