The following question concerns a remark made in the paper:
Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 (1991), Part 2.
Specifically, we want to show that the Taub-NUT metric on $\mathbb{S}^3 \times \mathbb{R}^+$ is a Ricci-flat Kähler metric using the Gibbons-Hawking ansatz.
Let $V : = 1 + 1/(2r) : \mathbb{R}^3 - \{ 0 \} \to \mathbb{R}^+$, and let $\pi : M_0 \to \mathbb{R}^3 - \{ 0 \}$ denote the $\mathbb{S}^1$-bundle of Chern class $-1$, which we equip with the connection of curvature $\textbf{F} = \ast dV$. Let $\omega$ denote the connection form of this connection, and let $ds^2$ denote the Euclidean metric on $\mathbb{R}^3$. Then $$g = V \pi^{\ast} ds^2 + V^{-1} \omega^2,$$ gives a metric on $M_0$ called the Taub-NUT metric.
Let now $\partial/\partial x$ be a unit vector on $\mathbb{R}^3$, and take some local trivialisation of $\pi : M_0 \to \mathbb{R}^3$ with vertical coordinate $t$, so that $\omega = dt + \vartheta$ for some $1$-form $\vartheta$ on $\mathbb{R}^3$ satisfying $d \vartheta = \ast dV$ which represents the connection in this gauge.
Then our complex structure $J$ is given on $1$-forms by $$dx \mapsto V^{-1}(dt + \vartheta), \hspace{0.5cm} dy \mapsto dz,$$ which is indeed integrable because \begin{eqnarray*} d[V dx + i(dt + \vartheta)] &=& [(-V_y + i V_z) dx - i V_x dz] \wedge (dy + i dz), \end{eqnarray*}
showing that the differential ideal generated by $dy + i dz$ and $V dx + i(dt + i\vartheta)$ is closed.
Q: Does someone have a reference for this definition of integrability?
