Schwarzschild time-translation Killing field in Kruskal coordinates ($K=(-U{\partial }_U+V{\partial }_V)/(4m)$)

For null coordinates normalized by $UV=(2m-r)e^{r/(2m)}$, the displayed field agrees with exterior Schwarzschild time translation. Its flow rescales $U$ and $V$ inversely, preserving their product, the radius and $dUdV$. It therefore preserves the extended metric across both horizons. Its squared norm is $1-2m/r$: it is timelike in both exterior regions, spacelike in the black-hole and white-hole interiors, and nonzero null on the horizons away from the bifurcation two-sphere. At that sphere it vanishes. The zero vector should not be mistaken for a nonzero null generator. This gives an explicit symmetry that survives where the original time coordinate ceases to be regular.