Let $X=\mathbb{C}^{3}-\{1+z_{1}z_{2}z_{3}=0\}$ with canonical coordinates $z_{1},z_{2},z_{3}$ and the standard symplectic structure. Consider the $T^{2}$-action on $X$ given by $(z_{1},z_{2},z_{3})\mapsto(e^{i\theta_{1}}z_{1},e^{i\theta_{2}}z_{2},e^{-i(\theta_{1}+\theta_{2})}z_{3})$. We obtain $f:X\rightarrow\mathbb{R}^{3}$ given by $f=(f_{1},f_{2},f_{3})$ where

$f_{1}=\log|1+z_{1}z_{2}z_{3}|$,

$f_{2}=|z_{1}|^{2}-|z_{2}|^{2}$,

$f_{3}=|z_{1}|^{2}-|z_{3}|^{2}$.

It is straightforward to check that the above functions Poisson commute, hence the fibres of $f$ are Lagrangian. It follows that $f$ is modeled on Example 4.3 near $\Crit(f)$. In particular, the discriminant locus is a cone over three points which coincides with the one in Example 4.3. This example has the topology of a positive fibration.