Concerning complete orientable minimal surfaces with finite total curvature in Euclidean three-space, we show for any positive genus the existence of noncongruent examples having the same symmetry group and conformal type.
"Noncongruent minimal surfaces with the same symmetries and conformal structure." Tohoku Math. J. (2) 56 (2) 237 - 254, 2004. https://doi.org/10.2748/tmj/1113246552