López' Klein bottle with one end and its generalization by López and Martín

Weierstrass Data (Double Cover of the Surface)

\[ \overline{M}=\left\{(z,w)\in(\mathbb{C}\cup\{\infty\})^2\;;\;w^{k+1}=\frac{z(z-r)}{rz+1}\right\}, \] where $k$ is positive integer and $r\in (-1,0)$ is a constant. \[ M=\overline{M}\setminus\{(0,0),\,(\infty ,\infty)\}, \] \[ g=w^k\frac{z-1}{z+1},\qquad \eta = i\frac{(z+1)^2}{z^2w^k}dz. \] For given $k$, we can choose $r$ so that $f$ is single valued on $M$.