The surfaces on this page are translation invariant of genus 2 and have the coordinate planes as symmetry planes.
The existence of this 2-parameter family can be established near their nodal limits as a consequence of Kevin Li’s work.
Below is an image far away from the nodal limit. The general behavior of these surfaces near their limits is not explored.
The special case that four of the ends are horizontal leaves us with a slightly simpler to describe 1-parameter family.
Mathematica Notebook and CDF (horizontal case)
PoVRay Sources (horizontal case)