A compact Riemann surface of genus $g$ is called an M-surface if
it admits an anti-conformal involution that fixes $g+1$ simple
closed curves, the maximum number by Harnack's Theorem. Underlying
every map on an orientable surface there is a Riemann surface and
so the conclusions of Harnack's theorem still apply. Here we show
that for each genus $g ϯ 1$ there is a unique M-surface of genus $g$
that underlies a regular map, and we prove a similar result for
Riemann surfaces admitting anti-conformal involutions that fix $g$
curves.