Let $(b_{1},\ldots,b_{n})$ be the standard coordinates on $\mathbb{R}^{n}$. Let $(U,\Gamma)$ be a pair of subsets of $\mathbb{R}^{n}$ diffeomorphic to $(D^{n},D^{n-1})$ and $\Gamma=U\cap\{b_{1}=0\}$. Define $U^{+}=U\cap\{b_{1}\geq 0\}$ and $U^{-}=U\cap\{b_{1}\leq 0\}$. Consider the lattice $\Lambda=\spn\langle db_{1},\ldots,db_{n}\rangle_{\mathbb{Z}}$ and form the symplectic manifold $T^{\ast}U/\penalty\Lambda$. Denote by $\pi$ the standard projection onto $U$. Let $Z=\pi^{-1}(\Gamma)$ and $\bar{Z}=Z/\penalty S^{1}$, where the $S^{1}$ action is the one generated by $db_{1}$. Suppose there is an open neighborhood $V\subseteq T^{\ast}U/\penalty\Lambda$ of $Z$ and a map $u:V\rightarrow\mathbb{R}^{n}$ which is a proper, smooth, $S^{1}$-invariant Lagrangian submersion with components $(u_{1},\ldots,u_{n})$ such that $u|_{Z}=\pi$ and $u_{1}=b_{1}$. Now define the following subsets of $T^{\ast}U/\penalty\Lambda$,

$$Y^{+}:=\pi^{-1}(U^{+}),\quad Y:=Y^{+}\cup V,\quad Y^{-}:=Y\cap\pi^{-1}(U^{-})$$

and define the map $f_{u}:Y\rightarrow\mathbb{R}^{n}$ by

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

(50)

Clearly $f_{u}:Y\rightarrow\mathbb{R}^{n}$ is a stitched fibration. Denote $B_{u}:=f_{u}(Y)$. The zero section $\sigma_{0}$ of $\pi$ is, perhaps after a change of coordinates in the base, a section of $f_{u}$. Let $\gamma_{0}$ be the basis of $H_{1}(Y,\mathbb{Z})$ induced by $\Lambda$. We call the stitched fibration $\mathcal{F}_{u}=(Y,B_{u},f_{u},\sigma_{0},\gamma_{0})$ a normal form.