  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.
