Consider $\mathbb{R}^{4}$ with standard coordinates $(x_{1},x_{2},y_{1},y_{2})$ and let $D^{4}\subseteq\mathbb{R}^{4}$. Let $D^{1}\times S^{1}$ have coordinates $(r,\theta)$. Define $V=D^{4}\times D^{1}\times S^{1}$ with the standard symplectic structure and $F(x_{i},y_{i},r,\theta)=(b_{1},b_{2},b_{3})$ where

$$\begin{array}[]{lll}b_{1}=x_{1}y_{1}+x_{2}y_{2},&b_{2}=x_{1}y_{2}-x_{2}y_{1},&b_{3}=r_{3}.\end{array}$$

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The reader may verify that $\mu=(b_{2},b_{3})$ is the moment map of a Hamiltonian action of $T^{2}$ and that $F$ is a $T^{2}$ invariant Lagrangian fibration of $V$ over $D^{2}\times D^{1}$. The singular fibres are homeomorphic to $\mathbb{R}\times S^{1}\times S^{1}$ after $\{p\}\times S^{1}\times S^{1}$ is collapsed to $\{p\}\times S^{1}$.

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

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

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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].

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.

Let $X^{\prime}=\mathbb{C}^{2}-\{z_{1}z_{2}-1=0\}$ and let $X=X^{\prime}\times\mathbb{C}^{\ast}$ with the standard symplectic structure. Define $f:X\rightarrow\mathbb{R}^{3}$ by $f=(f_{1},f_{2},f_{3})$ where

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

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

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

Again, these functions Poisson commute, hence $f$ is Lagrangian. The singular fibres of $f$ are lying over $\Delta=\{(0,r,0)\mid r\in\mathbb{R}\}$. The reader may verify that the above gives a generic-singular fibration.