This translation invariant surface is one of several singly periodic versions of the Costa surface. It has genus 1 in the quotient, two annular and two catenoidal ends.
It shares with the Costa surface the reflectional symmetries. The period problem is 2-dimensional and leaves a 1-parameter family. At one end one obtains the standard Costa surface.
I think that the other limit is a translation-invariant Costa-Scherk surface.