Suppose we have a Gibbons-Hawking solution on $R^{n}\times\mathbb{C}\setminus\Pi(\tau)\times\{0\}$, that is, a positive definite matrix function locally given by $V^{ij}=\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j}}$ such that $W:=\det V^{ij}=-4\frac{\partial^{2}\Phi}{\partial\eta\partial\bar{\eta}}$, which satisfy the distributional equation (10) in $R^{n}\times\mathbb{C}$ and, in addition, $\int\limits_{0}^{\infty}V^{ij}(u,\eta,\bar{\eta})du_{i}=\infty$. Then the total space of the torus bundle $\pi:M\to R^{n}\times\mathbb{C}\setminus\Pi(\tau)\times\{0\}$ can be compactified to the fibration $\bar{\pi}:\bar{M}\to R^{n}\times\mathbb{C}$ such that $\bar{M}$ is biholomorphic (in the orbifold sense) to $X_{\mathcal{T}}$ in a manner which respects the map

$$\eta:X_{\mathcal{T}}\to\mathbb{C}.$$

In particular, such solution defines a complete Ricci-flat Kähler metric on the orbifold $X_{\mathcal{T}}$ with the standard holomorphic volume form $\Omega_{\tau}$.