Let $(B,\Delta,\mathscr{A})$ be a simple affine 3-manifold with singularities and let $p,p^{\prime}\in\Delta$ be points connected by an edge $J$. Suppose there are disjoint neighborhoods $V$ and $V^{\prime}$ of $p$ and $p^{\prime}$ respectively and a neighborhood $W$ of $J$, with $W\cap\Delta=J$, such that the following conditions hold

• (i)

  if $\tilde{B}=B_{0}\cup(V\cup V^{\prime})$, there exists a Lagrangian fibration $\mathcal{F}=(X,\omega,f,\tilde{B})$ and a commuting diagram

  $$\begin{CD}X(B_{0},\mathscr{A})@>{\Psi}>{}>X\\ @V{f_{0}}V{}V@V{}V{f}V\\ B_{0}@>{\iota}>{}>\tilde{B}\end{CD}$$

  where $\Psi$ is a symplectomorphism and $\iota$ the inclusion.

• (ii)

  $\mathcal{F}|_{W\cap V}$ and $\mathcal{F}|_{W\cap V^{\prime}}$ are generic-singular fibrations.

Then, if we let $\tilde{B}^{\prime}=\tilde{B}\cup W$, there exists a Lagrangian fibration $\mathcal{F}^{\prime}=(X^{\prime},\omega^{\prime},f^{\prime},\tilde{B}^{\prime})$ and a commuting diagram

$$\begin{CD}X(B_{0},\mathscr{A})@>{\Psi^{\prime}}>{}>X^{\prime}\\ @V{f_{0}}V{}V@V{}V{f^{\prime}}V\\ B_{0}@>{\iota}>{}>\tilde{B}^{\prime}\end{CD}$$

where $\Psi^{\prime}$ is also a symplectomorphism.