Hermann Karcher described in 1988 a surprising deformation of the fully symmetric Scherk surfaces into screw motion invariant surfaces, the first new examples since the helicoid.
This can be done for any number 2k (k≥2) number of ends. The Weierstraß representation becomes multivalued on the 2k-punctured quotient spheres:
Here determines the screw motion angle, and R needs to be determined numerically to close a single period.
When the screw motion parameter a reaches its upper limit 1/k, the surfaces converge to a horizontal foliation of R³ by planes with singular vertical lines at the 2k-th roots of unity.
It is still an open question whether these screw-motion invariant Karcher-Scherk surfaces together with the helicoid are the only embedded, screw-motion invariant minimal surfaces with genus 0 in the quotient.
H. Karcher: Embedded Minimal Surfaces Derived from Scherk’s Examples, manuscripta math. 62, 83-114 (1988).