Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms

We study the symmetric Riemann surfaces for which the group of orientation preserving automorphisms acts without fixed points. We show that any finite group can give rise to such an action, determine the maximal number of non-conjugate symmetries for such surfaces and find a sharp upper bound on maximal total number of ovals for a set of k symmetries with ovals. We also solve the minimal genus problem for dihedral groups acting on the surfaces described above, for odd genera.