The Ernst equation and ergosurfaces
  We show that analytic solutions $\mcE$ of the Ernst equation with non-empty zero-level-set of $\Re \mcE$ lead to smooth ergosurfaces in space-time. In fact, the space-time metric is smooth near a "Ernst ergosurface" $E_f$ if and only if $\mcE$ is smooth near $E_f$ and does not have zeros of infinite order there. 
Piotr T. Chrusciel, Gert-Martin Greuel, Reinhard Meinel, Sebastian J.   Szybka