Let $Z$ be the seam of $\mathcal{F}$, $\omega_{red}$ the reduced symplectic form on $\bar{Z}$ and $\bar{f}:\bar{Z}\rightarrow\Gamma$ the reduced fibration. Using the coisotropic embedding theorem we can assume w.l.o.g. that $X=\mathbb{R}\times S^{1}\times\bar{Z}$ with symplectic form $\omega=\omega_{red}+ds\wedge dt$, where $(t,s)$ are coordinates on $\mathbb{R}\times S^{1}$ and the projection onto $\mathbb{R}$ is the moment map $\mu$. On $X$, we can define an “auxiliary” smooth Lagrangian fibration given by

$$\tilde{\pi}(t,s,p)=(t,\bar{f}(p)).$$

Fix a basis $\gamma$ of $H_{1}(X,\mathbb{Z})\cong H_{1}(S^{1}\times\bar{Z},\mathbb{Z})$ and a smooth Lagrangian section of $\tilde{\pi}$. The action-angle coordinates of $\tilde{\pi}$ with respect to $\gamma$ and $\sigma$ induce a $C^{\infty}$ symplectomorphism

$$T^{\ast}U/\penalty\Lambda\cong X$$

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for some open neighborhood $U$ of $0\in\mathbb{R}^{n}$ with action coordinates $(b_{1},\ldots,b_{n})$. The angle coordinates are $(y_{1},\ldots,y_{n})$. In these coordinates $Z=\{b_{1}=0\}$ and $\Gamma=U\cap\{b_{1}=0\}$. While $f$ becomes:

$$f=\begin{cases}u^{+}\quad\text{on}\ X^{+};\\ u^{-}\quad\text{on}\ X^{-},\end{cases}$$

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where $u^{\pm}$ correspond to $f^{\pm}$. It follows that $u^{+}|_{Z}=u^{-}|_{Z}=\pi|_{Z}$.

One can show that $u^{+}$ can be extended as a smooth proper Lagrangian fibration a little bit beyond $X^{+}$, i.e. we can find a smooth proper Lagrangian fibration $\tilde{u}^{+}$ defined on a set $X^{+}\cup V$, where $V$ is some open neighborhood of $Z$, such that $\tilde{u}^{+}|_{X^{+}}=u^{+}$. For the details of this extension see [2], Proposition 6.3. To put $f$ in normal form, we consider the action-angle coordinates associated to $\tilde{u}^{+}$ with section $\sigma$ and basis $\gamma$ of $H_{1}(X,\mathbb{Z})$ as above. In these coordinates, $X^{+}\cup V$ becomes $T^{\ast}U/\Lambda$ and $\tilde{u}^{+}$ becomes the projection $\pi$. Again in action-angle coordinates of $\tilde{u}^{+}$, a Lagrangian extension $\tilde{u}^{-}$ of $u^{-}$, becomes $(W,u)\in\mathscr{U}_{Z}$ for some $W\subseteq T^{\ast}U/\penalty\Lambda$ and some Lagrangian fibration $u$. Then we simply define $Y^{+}=T^{*}U^{+}/\Lambda$, $Y=Y^{+}\cup W$, $Y^{-}=Y\cap\pi^{-1}(U^{-})$ and

$$f_{u}=\begin{cases}u\quad\text{on}\ Y^{-},\\ \pi\quad\text{on}\ Y^{+}.\end{cases}$$

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