These surfaces are straightforward variations of the Chen-Gackstatter surface, which is the case n=2.
Their symmetries are generated by reflections at n vertical planes and rotations about n horizontal lines. The genus is n-1, and there is only one end.
The Weierstraß data are
and can be integrated in terms of hypergeometric functions.
Resources
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 