The Enneper-Weierstrass representation of the rotated S’-S” surface of Alan Schoen allows the introduction of an additional parameter.
For a long time I thought of this surface as the CLP surface with an additional inward handle, as the image above suggests. Different assemblies show that it is nothing but a deformation of S’-S”. Limits include 8-ended Karcher-Scherk surfaces, Rossman-Thayer-Wohlgemuth surfaces, and nodal limits.
The divisor of the squared Gauss map is shown below. Abel tells us that a+b=c, and we still need to solve a 1-dimensional period problem.