Let $U\subset\mathbb{R}^{2}$ be an annulus as above with coordinates $(b_{1},b_{2})$. Let $\bar{Z}_{d}=T^{\ast}\Gamma_{d}\,/\,\langle db_{2}\rangle_{\mathbb{Z}}$ and $\bar{Z}_{u}=T^{\ast}\Gamma_{u}\,/\,\langle db_{2}\rangle_{\mathbb{Z}}$ with projections $\bar{\pi}^{d}$ and $\bar{\pi}^{u}$ and bundles $\mathfrak{L}_{d}=\ker\bar{\pi}^{d}_{\ast}$ and $\mathfrak{L}_{u}=\ker\bar{\pi}^{u}_{\ast}$ respectively. Given an integer $m$ and sequences $\ell^{d}=\{\ell_{k}^{d}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{d}}$ and $\ell^{u}=\{\ell_{k}^{u}\}_{k\in\mathbb{N}}\in\mathscr{L}_{\bar{Z}_{u}}$ such that

$$\int_{[db_{2}]}\ell_{1}^{u}=0\ \ \text{and}\ \ \int_{[db_{2}]}\ell_{1}^{d}=m,$$

there exists a smooth symplectic manifold $(X,\omega)$ and a stitched Lagrangian fibration $f:X\rightarrow U$ having monodromy (57) with respect to some basis $\gamma=\{\gamma_{1},\gamma_{2}\}$ of $H_{1}(f^{-1}(U-\Gamma_{d}),\mathbb{Z})$ and satisfying the following properties:

• (i)

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

• (ii)

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

• (iii)

  there is a Lagrangian section $\sigma$ of $f$, such that $(\bar{Z}_{u},\ell^{u})$ and $(\bar{Z}_{d},\,\ell^{d})$ are the invariants of $(f^{-1}(U-\Gamma_{d}),\,f,\,U-\Gamma_{d},\,\sigma,\,\gamma)$ and $(f^{-1}(U-\Gamma_{u}),\,f,\,U-\Gamma_{u},\,\sigma,\,j_{+}(\gamma))$ respectively.

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