Double Enneper of genus $k$
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$k=2$, $a=-0.1$, $b\approx 0.288741$.
Weierstrass Data
\[ \overline{M}=\big\{(z,w)\in(\mathbb{C}\cup\{\infty\})^2\;;\;w^{k+1}=z^k(z-b)(1-bz)^k\big\}, \] where $k$ is positive integer and $a$, $b$ are constant so that \( -1 < -a < 0 < b < 1 \). \[ M=\overline{M}\setminus\{(0,0),\,(\infty ,\infty)\}, \] \[ g=\frac{(az-1)w}{(z-a)(z-b)},\qquad \eta = \frac{(z-a)^2(z-b)}{z^2w}dz. \] For given $k$ and $a$, we can choose $b$ so that $f$ is single valued on $M$.
