Let S be a compact Riemann surface without boundary. A symmetry of S is an anti-conformal, involutary automorphism. Its fixed point set is a disjoint union of circles, each of which is called an oval. A method is presented for counting the ovals of a symmetry when S admits a large group G of automorphisms. The method involves only calculations in G, based on the geometric description of S/G, and the knowledge of the action of the symmetry on G.
Broughton, Sean A., "Counting Ovals on a Symmetric Riemann Surface" (1997). Mathematical Sciences Technical Reports (MSTR). 68.