Let $U\subset\mathbb{R}^{3}$, $\Gamma_{c}$, $\Gamma_{d}$ and $\Gamma_{e}$ be as in Example 6.18 and let $(b_{1},b_{2},b_{3})$ be coordinates on $U$. Define $\bar{Z}_{c}=T^{\ast}\Gamma_{c}\,/\,\langle db_{2},db_{3}\rangle_{\mathbb{Z}}$, $\bar{Z}_{d}=T^{\ast}\Gamma_{d}\,/\,\langle db_{2},db_{3}\rangle_{\mathbb{Z}}$ and $\bar{Z}_{e}=T^{\ast}\Gamma_{e}\,/\,\langle db_{2},db_{3}\rangle_{\mathbb{Z}}$ with projections $\bar{\pi}^{c}$, $\bar{\pi}^{d}$, $\bar{\pi}^{e}$ and bundles $\mathfrak{L}_{c}=\ker\bar{\pi}^{c}_{\ast}$, $\mathfrak{L}_{d}=\ker\bar{\pi}^{d}_{\ast}$, $\mathfrak{L}_{e}=\ker\bar{\pi}^{e}_{\ast}$. Suppose we are given integers $m_{1}$, $m_{2}$ and sequences $\ell^{c}=\{\ell_{k}^{c}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{c}}$, $\ell^{d}=\{\ell_{k}^{d}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{d}}$ and $\ell^{e}=\{\ell_{k}^{e}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{e}}$ satisfying

Then there exists a smooth symplectic manifold $(X,\omega)$ and a stitched Lagrangian fibration $f:X\rightarrow U$ having the same monodromy of Example 6.18 with respect to some basis $\gamma=\{\gamma_{1},\gamma_{2},\gamma_{3}\}$ of $H_{1}(f^{-1}(U-(\Gamma_{d}\cup\Gamma_{e})),\mathbb{Z})$ and satisfying the following properties:

• (i)

  the coordinates $(b_{1},b_{2},b_{3})$ are action coordinates of $f$ with moment map $f^{\ast}b_{1}$;

• (ii)

  the periods $\{db_{1},db_{2},db_{3}\}$, restricted to $U^{\pm}$ correspond to the basis $\gamma$;

• (iii)

  there is a Lagrangian section $\sigma$ of $f$, such that $(\bar{Z}_{c},\ell^{c})$, $(\bar{Z}_{d},\,\ell^{d})$ and $(\bar{Z}_{e},\,\ell^{e})$ are respectively the invariants of:

  $$(f|_{U-(\Gamma_{d}\cup\Gamma_{e})},\sigma,\,\gamma),\ (f|_{U-(\Gamma_{c}\cup\Gamma_{e})},\sigma,\,j_{+}(\gamma))\ \textrm{and}\ (f|_{U-(\Gamma_{c}\cup\Gamma_{d})},\sigma,\,j_{+}(\gamma)).$$

The fibration $(X,f,U)$ satisfying the above properties is unique up to fibre preserving symplectomorphism.