establish our conventions for the Kaluza-Klein monopole in 11-dimensions. A useful form of this metric is [@NOYY; @Yosh; @BG]

$$ds^2 = - dx_\parallel^2 + V dv d\overline v + V^{-1} \left|
\frac{dw}{w} - f dv \right|^2,$$

where $$f = \frac{x_9 + r}{2vr},$$ correcting a small typographic error in [@NOYY; @BG]. The complex coordinates $v$ and $w$ define one of the complex structures on the Euclidean Taub-Nut space. They are related to the ten-dimensional coordinates through $$\begin{aligned}
v &=& x_7 + i x_8 \cr
w &=& e^{-(x_9+ix_{10})} \left( - x_9 + \sqrt{x_9^2 + |v|^2} \right)^{1/2},\end{aligned}$$ and Kaluza-Klein reduction takes place along the Killing field $\partial_{x_{10}}$. Such coordinates are smooth so long as $v \neq 0$ or $x_9 < 0$. Note that $x_{10}$ ranges over $[0,2\pi]$ consistent with our setting $R_{10}=1$. In addition, a careful check will show that this space has a $Z_2$ symmetry of the form $(w,v) \rightarrow (\frac{v}{w}, v)$.

A holomorphic curve that reduces to the surface $x_9=0$ surface can be found by noticing that the symmetry $(w,v) \rightarrow (\frac{v}{w},v)$ changes the sign of $x_9$, so that any surface which is invariant under this symmetry must lie at $x_9=0$. The surface $w^2 = v$ is invariant in this way, 