We refer the reader to [2]Theorem 6.12 for the details. Roughly, one starts with $U^{+}$ and $U^{-}$ regarded as disjoint sets. These give two disjoint pieces $X^{\pm}=T^{\ast}U^{\pm}/\penalty\Lambda^{\pm}$, where $\Lambda^{\pm}=\langle db_{1},\ldots,db_{n}\rangle_{\mathbb{Z}}$. Let $Z^{\pm}=\partial X^{\pm}$. On $X^{+}$ we have Hamiltonian vector fields $\eta_{1}=\partial_{b_{1}}$ and $\eta^{+}_{j}=\partial_{b_{j}}$ for $j=2,\ldots,n$. We can also define vector fields on $Z^{+}$:

$$\eta^{-}_{j}=\eta^{+}_{j}-a_{j}\eta_{1}$$

where $(a_{2},\ldots,a_{n})$ are the coefficients of $\ell_{1}$. One can (topologically) glue $X^{+}$ and $X^{-}$ using a map $Q:Z^{-}\rightarrow Z^{+}$ defined in terms of the $\mathbb{R}^{n}$ action induced by the flows of $\eta_{j}^{-}$. Intuitively, $Q$ identifies the fibres inside each of the two halves $Z^{-}$ and $Z^{+}$ after the fibres inside $Z^{-}$ have been twisted by iteratively flowing in the direction of $\eta_{1},\eta^{-}_{2},\ldots,\eta^{-}_{n}$. The integrality condition (56) guarantees that (ii) is satisfied. One can extend $Q$ to give a smooth symplectomorphism $\tilde{Q}$ between open neighborhoods of $Z^{\pm}$. For this one needs to consider invariants $\ell_{k}$, for $k>1$. The choice of $\tilde{Q}$ is determined by $\{\ell_{k}\}$. This gluing gives a smooth symplectic manifold $(X,\omega)$ and a stitched fibration $f:X\rightarrow U$, which by construction is such that $\inv(\mathcal{F})=(\bar{Z}_{\text{nor}},\ell)$. ∎