As part of his thesis, Peter Connor discovered a new doubly periodic surface of genus 2 that has fewer symmetries than expected.

This surface comes in a 1-parameter family which moves the little catenoidal neck up and down:

The left limit appears to be a Karcher-Scherk surface, glued to a standard Scherk surface. The right surface degenerates when the catenoidal neck approaches the middle into two standard Scherk surfaces which are glued using catenoidal necks. Note that the changing normal between top and bottom ends prevents a horizontal symmetry plane. 

 All this has only been established only numerically. The period problem is 2-dimensional, so should be manageable with some effort. It would be more worthwhile to prove existence by regenerating them from the nodal limits.

Even more interesting would be to prove or disprove that these surfaces can be deformed into one of two other known examples of doubly periodic genus 2 surface with parallel ends, the genus 2 Wei surfaces or the RTW M1+ surface.

Resources

Mathematica Notebook or CDF

PoVRay Sources