  It is a well-known fact that the Schwarzschild spacetime admits a maximal
spacetime extension in null coordinates which extends the exterior
Schwarzschild region past the Killing horizon, called the Kruskal-Szekeres
extension. This method of extending the Schwarzschild spacetime was later
generalized by Brill-Hayward to a class of spacetimes of "profile $h$" across
non-degenerate Killing horizons. Circumventing analytical subtleties in their
approach, we reconfirm this fact by reformulating the problem as an ODE, and
showing that the ODE admits a solution if and only if the naturally arising
Killing horizon is non-degenerate. Notably, this approach lends itself to
discussing regularity across the horizon for non-smooth metrics.
  We will discuss applications to the study of photon surfaces, extending
results by Cederbaum-Galloway and Cederbaum-Jahns-Vi\v{c}\'{a}nek-Mart\'{i}nez
beyond the Killing horizon. In particular, our analysis asserts that photon
surfaces approaching the Killing horizon must necessarily cross it.
