By Proposition 3.1, the Calabi space $\mathcal{C}$ is diffeomorphic to $\Nil^{3}_{b}\times(0,\infty)$ in such a way that the Calabi metric $g_{\mathcal{C}}$ becomes the Gibbons-Hawking metric (2.19). Ignoring this diffeomorphism, we have a $2$-form triple ${\bm{\phi}}$ and a $1$-form triple ${\bm{\psi}}$ such that $\partial_{z}\,\lrcorner\,{\bm{\phi}}=0$, $\partial_{z}\,\lrcorner\,{\bm{\psi}}=0$, and

(3.35)

$$\Phi^{*}\bm{\omega}_{TY}-\bm{\omega}_{\mathcal{C}}=\bm{\phi}+dz\wedge\bm{\psi}.$$

Since $\bm{\omega}_{TY},\bm{\omega}_{\mathcal{C}}$ are closed, it follows that

(3.36)

$$d_{\Nil^{3}_{b}}\bm{\phi}=0,\;\,\partial_{z}\bm{\phi}-d_{\Nil^{3}_{b}}\bm{\psi}=0.$$

Now we define the $1$-form triple

(3.37)

$$\bm{a}\equiv-\int_{z}^{\infty}{\bm{\psi}}\,dz.$$

Thanks to (3.32), this integral exists and satisfies (3.34). Note that this is not completely obvious because the $1$-form basis $dx,dy,dt-xdy$ on ${\rm Nil}^{3}_{b}$ is not parallel with respect to $g_{\mathcal{C}}$. However, this effect is absorbed by the $\epsilon$ in (3.34) because all error terms are at worst polynomial in $z$. Property (3.33) now follows in a standard manner by using (3.36). ∎