Given a domain ${B}$ in $\mathbb{R}^{n}\times(\mathbb{C}^{*})^{l}$ a $(\sigma,\tau)$-type solution to the Gibbons-Hawking ansatz in ${B}$ are two positive definite matrix functions – a real $V^{ij}$ and a hermitian $W^{pq}$ – on $B^{\circ}$ locally given by a potential:

$$V^{ij}=\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j}},\quad W^{pq}=-4\frac{\partial^{2}\Phi}{\partial\eta_{p}\partial\bar{\eta}_{q}},\qquad 1\leq i,j\leq n,\quad n+1\leq p,q\leq n+l,$$

such that $\det V^{ij}=\det W^{pq}$ and the distributional equation

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$$\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}W^{pq}}{\partial u_{i}\partial u_{j}}+4\frac{\partial^{2}V^{ij}}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)du_{i}\wedge d\eta_{p}\wedge d\bar{\eta}_{q}=\gamma^{j}_{\tau}(u)\wedge\Gamma_{\sigma}(\eta)$$

is satisfied in ${B}$.