All Chen-Gackstatter surfaces of genus g can be symmetrized, increasing both the genus and the winding number of the ends. In general, this requires solving a (g-1)-dimensional period problem. 

Besides the existence, not much is known. Do they converge for fixed symmetry and increasing genus to the symmetric Karcher-Scherk surfaces? Can one add vertical handles as in the toroidal Karcher-Scherk surfaces?

Resources

Mathematica Notebook and CDF

PoVRay Sources

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