The Wikipedia article about the Taub-NUT spacetime says that it was a first attempt in finding the Kerr solution. Since the Kerr spacetime is a stationary solution, meaning that it admits an asymptotically timelike Killing field (that is, near the future and past null infinities $\mathscr I^\pm$), I wonder if that is true also for the Taub-NUT spacetime.
In particular, what is its stationary limit (the so-called ergosphere)? Is it finite or infinite? Any ideas?