Unique continuation and extensions of Killing vectors at boundaries for stationary vacuum space-times

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.