The Ernst equation and ergosurfaces

We show that analytic solutions $\text{mcE}$ of the Ernst equation with non-empty zero-level-set of $\mathrm{\Re }\text{mcE}$ 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 $\text{mcE}$ is smooth near $E_f$ and does not have zeros of infinite order there.