Let $(U,\Gamma)$ be a pair of subsets of $\mathbb{R}^{2}\times\mathbb{R}$ diffeomorphic to $(D^{2}\times D^{1},D^{1}\times D^{1})$ with $\Gamma=U\cap\{b_{1}=0\}$. Let $\Delta=\{b_{1}=b_{2}=0\}$. Given $H\in C^{\infty}(U)$ denote by $H_{\Delta}$ the germ of $H$ along $\Delta$. Consider the integral lattice $\Lambda_{H}$ in $T^{\ast}U$ generated by:

$$\begin{array}[]{l}\lambda_{1}=2\pi db_{1},\\ \lambda_{2}=dH+\arg(b_{1}+ib_{2})db_{1}+\log|b_{1}+ib_{2}|db_{2},\\ \lambda_{3}=db_{3}.\end{array}$$

(64)

Let $(y_{1},y_{2},y_{3})$ denote the locally defined vertical coordinates on $T^{*}U$, which it is convenient to think of as $\Lambda_{H}$-periodic coordinates. For fixed positive $L\in\mathbb{R}$ consider the following subset of $T^{*}U$:

$$C_{L}=\{|y_{2}|<L\}$$

(65)

and denote $C_{L}(b)=T^{*}_{b}U\cap C_{L}$. If $U$ is a sufficiently small neighborhood of $\Delta$, we can assume that for every $b\in U$, $2L<|\log|b|+\partial_{b_{2}}H|$. Therefore the projection $T^{*}_{b}U\rightarrow T^{*}_{b}U/\Lambda_{H}$ maps $C_{L}(b)$ to a cylinder which closes up in the $y_{1}$ and $y_{3}$ direction but not in the $y_{2}$ direction. So let us think of $C_{L}(b)$ as this cylinder and define $J^{\circ}_{L}=\bigsqcup_{b\in U}C_{L}(b)$, which is an open subset of $T^{*}_{b}U/\Lambda_{H}$. The projection $\pi$ restricts to an open cylinder fibration:

$$\pi^{\circ}:J^{\circ}_{L}\rightarrow U.$$

Clearly there is an $S^{1}$ action on $J^{\circ}_{L}$ induced by $\lambda_{1}$, whose moment map is $b_{1}$. Let $Z^{\circ}_{L}=(\pi^{\circ})^{-1}(\Gamma)$ and let $\bar{Z}^{\circ}_{L}$ be the corresponding $S^{1}$ reduced space. Let $\bar{\pi}^{\circ}:\bar{Z}_{L}^{\circ}\rightarrow\Gamma$ be the reduced fibration. We denote the fibre of $\bar{\pi}^{\circ}$ by $\bar{C}_{L}(b)$.

For $L^{\prime}<L$, construct $J^{\circ}_{L^{\prime}}$, which is a cylinder fibration with shorter cylinders, and define its closure $K_{L^{\prime}}=\overline{J^{\circ}_{L^{\prime}}}$. Define the open set $E_{L,L^{\prime}}=J^{\circ}_{L}-K_{L^{\prime}}$, which we can think of as the union of the ends of the cylinders. Suppose now that we have an open neighborhood $V$ of $Z^{\circ}_{L}$ and a smooth $S^{1}$ invariant Lagrangian submersion $u:V\rightarrow\mathbb{R}^{3}$ with cylindrical fibres satisfying: $u|_{Z^{\circ}_{L}}=\pi^{\circ}$, $u|_{E_{L,L^{\prime}}}=\pi^{\circ}$ and $u_{1}=b_{1}$. Then we can define $Y^{+}_{L}=(\pi^{\circ})^{-1}(U^{+})$, $Y_{L}=Y^{+}_{L}\cup V$, $Y^{-}_{L}=Y_{L}\cap(\pi^{\circ})^{-1}(U^{-})$ and the piecewise smooth function $f^{\circ}_{u}:Y_{L}\rightarrow B_{u}\subseteq\mathbb{R}^{n}$ to be the map

$$f^{\circ}_{u}=\begin{cases}\pi^{\circ}\quad\text{on}\ Y^{+}_{L},\\ u\quad\text{on}\ Y^{-}_{L}.\end{cases}$$

(66)

Clearly, if we think of $Y_{L}$ as playing the role of $X^{\circ}$, $f^{\circ}_{u}:Y_{L}\rightarrow B_{u}$ is a Lagrangian fibration of type (62). Notice that the fibres of $f^{\circ}_{u}$ coincide with the fibres of $\pi^{\circ}$ inside $E_{L,L^{\prime}}$, in particular $f^{\circ}_{u}$ is smooth restricted to $E_{L,L^{\prime}}$. In some sense, the fibres of $f_{u}$ are straight towards their ends (cf. Figure 11).

We now compactify by adding the singularities. Let $J_{H}^{\#}=T^{\ast}U/\penalty\Lambda_{H}$ and let $\pi^{\#}:J^{\#}_{H}\rightarrow U$ be the Lagrangian fibration induced by the standard projection on $T^{\ast}U$. Clearly $J^{\circ}_{L}$ and therefore $Y_{L}$ are open subsets of $J_{H}^{\#}$. When $b\in\Delta$, the fibre $C(b)=(\pi^{\#})^{-1}(b)$ is an open cylinder, with ends at $+\infty$ and $-\infty$ in the $y_{2}$-direction, otherwise $C(b)$ is a torus. From the results in [1], $J_{H}^{\#}$ can be compactified to a symplectic manifold $X$ by adding the singularity at the ends of the cylinders $C(b)$ when $b\in\Delta$. The fibration $\pi^{\#}$ extends to a smooth fibration $f_{H}:X\rightarrow U$ of generic-singular type. The open subset $J^{\#}_{H}-K_{L^{\prime}}$ extends to an open neighborhood $E$ of the singular set $\Sigma$. The fibres of $f^{\circ}_{u}$ coincide with the fibres of $f_{H}$ toward their ends and therefore $f^{\circ}_{u}$ may be extended to make it coincide with $f_{H}$ on $E$. More precisely, define $\mathfrak{U}=f^{-1}_{H}(B_{u})\cap E$ and $Y=Y_{L}\cup\mathfrak{U}$. Now we can define

$$f_{u,H}=\begin{cases}f_{H}\quad\text{on}\ \mathfrak{U},\\ f^{\circ}_{u}\quad\text{on}\ Y_{L}.\end{cases}$$

(67)

Clearly $f_{u,H}:Y\rightarrow B_{u}$ is a well defined Lagrangian fibration satisfying Assumption 6.22. The zero section $\sigma_{0}$ of $\pi^{\circ}$ is, perhaps after a change of coordinates in the base, a section of $f_{u}$. If $F_{\bar{b}}$ is a smooth fibre of $f_{u,H}$, with $\bar{b}\in U^{+}$, let $\gamma_{0}$ be the basis of $H_{1}(F_{\bar{b}},\mathbb{Z})$ determined by $\lambda_{1},\lambda_{2},\lambda_{3}$. We call $\mathcal{F}_{u,H}=(Y,f_{H,u},\sigma_{0},\gamma_{0})$ a normal form of cylindrical type.

The set $Y_{L}\subset J^{\#}_{H}$ can be visualized in Figure 11 as the square with open top and bottom. The straight light-colored lines are the fibres of $\pi^{\#}$ and the fibres of $f^{\circ}_{u}:Y_{L}\rightarrow B_{u}$ are depicted as dark lines. The upper and lower rectangular regions represent the components of $E_{L,L^{\prime}}$.