The natural $S^{1}$-action on $\mathcal{C}$ obviously preserves ${\bm{\omega}}_{\mathcal{C}}$, so ${\bm{\omega}}_{\mathcal{C}}$ must be given by the Gibbons-Hawking construction. It suffices to determine the hyperkähler moment map $\pi=(\pi_{1},\pi_{2},\pi_{3})$ and the potential $V$. We already know that the $\omega_{\mathcal{C}}$-moment map is given by $\pi_{1}=(-{\log|\xi|_{h}^{2}})^{1/2}=z$, and it is easy to check that the $\Omega_{\mathcal{C}}$-moment map $\pi_{2}+i\pi_{3}$ equals the bundle projection $p:\mathcal{C}\to E$ followed by the Abel-Jacobi isomorphism $E\to\mathbb{C}/\text{\emph{periods}}$, $x\mapsto\int_{x_{0}}^{x}i\Omega_{E}$, for an arbitrary but fixed basepoint $x_{0}\in E$. Also, $V^{-1}$ is the norm-squared of the Killing field, so that

$$V^{-1}=(-{\log|\xi|^{2}_{h}})^{-1/2}=z^{-1}.$$

The Calabi construction provides us with an explicit realization $\mathfrak{M}=\mathcal{C}$ of the total space of the $S^{1}$-bundle and with a specific choice of connection form $\theta$ given by the Chern connection of $L$ with respect to $h$. The diffeomorphism equivalence to the model ${\rm Nil}_{b}^{3}(\epsilon,\tau)\times(0,\infty)$ with connection form $\theta_{b}$ (after rotating $\Omega_{\mathcal{C}}$ to $e^{i\alpha}\Omega_{\mathcal{C}}$ if necessary) follows from the discussion before Remark 2.4. ∎