Using Proposition 5.4 we can obtain the following piecewise smooth fibration:

$$f(z_{1},z_{2})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|\gamma(z_{1},z_{2})+1|\right).$$

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It is clearly well defined on $X=\{(z_{1},z_{2})\in\mathbb{C}^{2}\ |\ \gamma(z_{1},z_{2})+1\neq 0\}.$ Observe that $f$ is topologically conjugate to a focus-focus fibration, hence to Example 2.6. The only singular fibre is $f^{-1}(0)$ and it is a pinched torus. The fibration fails to be smooth on $\mu^{-1}(0)$. This example consists of the union of two smooth Lagrangian fibrations meeting along the “stitch”, $\mu^{-1}(0)$. We study this kind of piecewise smoothness in detail in §6.