(Taub-NUT type metric on $\mathbb{C}^{3}$, cf. Chapter 2) There is a family of Calabi-Yau metrics on $(\mathbb{C}^{3}_{z_{0},z_{1},z_{2}},-dz_{0}\wedge dz_{1}\wedge dz_{2})$ invariant under the diagonal $T^{2}$-action, which are parametrised by positive definite rank 2 real symmetric matrices $(a_{ij})$. The base of the $T^{2}$-fibration is $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$ and the discriminant locus is the trivalent graph

$$\mathfrak{D}=\{\mu_{2}\geq 0,\mu_{1}=0,\eta=0\}\cup\{\mu_{1}\geq 0,\mu_{2}=0,\eta=0\}\cup\{\mu_{1}=\mu_{2}\leq 0,\eta=0\}.$$

The tangent cone at infinity is the Euclidean $\mathbb{R}^{4}$. Near spatial infinity suitably away from $\mathfrak{D}$, the metric is approximately a flat $T^{2}$ fibration over an open subset of Euclidean $\mathbb{R}^{4}$, such that the metric on the $T^{2}$-fibres is asymptotically given by the inverse matrix $(a^{ij})$ in distinguished coordinates. The metric transverse to $\mathfrak{D}$ is modelled on a fibration by Taub-NUT metrics.