These surfaces are straightforward variations of  the Chen-Gackstatter surface, which is the case n=2.cg-3

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.cg-7

The Weierstraß data are

G(z) = z^{\frac{1}{n}-1} \left(1-z^2\right)^{1-\frac{1}{n}} \quad\text{and}\quad dh = dz

and can be integrated in terms of hypergeometric functions.

Resources

Mathematica Notebook

PoVRay Sources