One can view this surface as a translation invariant version of the Finite Riemann surface.
The quotient under the horizontal translation has two annular and two catenoidal ends, all with vertical limiting normal. The López-Ros factor changes the growth rates of the catenoidal ends.
Other placements of the catenoidal necks are possible:
Their locations are controlled by elementary versions of Traizet’s balance equation. In a configuration with one bottom catenoidal neck and n top catenoidal necks of equal neck size, arranged in the line, these locations are determined by the roots of Legendre polynomials. The details have been worked out by Peter Connor.