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

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

(14)

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 $\{p\}\times S^{1}$.