This symmetrization of the Finite Riemann minimal surface has one planar end and 2n catenoidal ends that are more or less tilted.

Very little is known about such planes with added catenoidal necks. By the Lopez-Ros theorem, they can never be embedded. But it would be amusing to study their moduli space for a small number (say two or three) of catenoidal necks.

Resources:

Mathematica Notebook

Mathematica CDF

PoVRay sources