Consider $\mathbb{C}^{3}$ with canonical coordinates $z_{1},z_{2},z_{3}$. Define $F(z)=(b_{1},b_{2},b_{3})$, where

$$\begin{array}[]{lll}b_{1}=\im z_{1}z_{2}z_{3},&b_{2}=|z_{1}|^{2}-|z_{2}|^{2},&b_{3}=|z_{1}|^{2}-|z_{3}|^{2}.\end{array}$$

(15)

Here $\mu(z_{1},z_{2},z_{3})=(b_{2},b_{3})$ is the moment map of a $T^{2}$-action, furthermore the above functions Poisson commute, so the fibres of $F$ are Lagrangian. The critical locus of $F$ is $\Crit(F)=\bigcup_{ij}\{z_{i}=z_{j}=0\}$ and its discriminant locus is $\Delta=\{b_{1}=0,b_{2}=b_{3}\geq 0\}\cup\{b_{1}=b_{2}=0,b_{3}\leq 0\}\cup\{b_{1}=b_{3}=0,b_{2}\leq 0\}$, i.e. a cone over three points with vertex at $0\in\mathbb{R}^{3}$. The regular fibres are homeomorphic to $\mathbb{R}\times T^{2}$. The singular fibre over $0\in\Delta$ is homeomorphic to $\mathbb{R}\times T^{2}$ after $\{p\}\times T^{2}$ is collapsed to $p\in\mathbb{R}$. All the other singular fibres are homeomorphic to $\mathbb{R}\times T^{2}$ after a two cycle $\{p\}\times T^{2}\subset\mathbb{R}\times T^{2}$ is collapsed to a circle. This is one of the examples of special Lagrangian fibrations by Harvey and Lawson [19].