Let $(X,\omega)$ be a $6$-dimensional symplectic manifold and $B\subseteq\mathbb{R}^{3}$ an open subset. Let $f:X\rightarrow B$ be a piecewise smooth Lagrangian fibration. $\mathcal{F}=(X,\omega,f,B)$ is called a Lagrangian negative fibration if it satisfies the following properties:

• (i)

  $\mathcal{F}$ is topologically conjugate to the alternative negative fibration of Example 2.9.

• (ii)

  there exists a submanifold with boundary $D\subset B$, homeomorphic to a closed disc in $\mathbb{R}^{2}$, such that $\Delta\cap(B-D)$ consists of three one dimensional disjoint segments (the legs of $\Delta$) and $f$ is smooth when restricted to $X-f^{-1}(D)$;

• (iii)

  let $B_{0}=B-(D\cup\Delta)$, $X_{0}=f^{-1}(B_{0})$ and $f_{0}=f|_{X_{0}}$. Let $(B_{0},\mathscr{A})$ be the integral affine manifold induced by the Lagrangian $T^{3}$ bundle $\mathcal{F}_{0}=(X_{0},f_{0},B_{0})$. For some choice of model of negative vertex $(\mathbb{R}^{3},\Delta_{\tau},\mathscr{A}_{\tau})$ as given in Example 3.13, there exist an open neighborhood $U\subseteq\mathbb{R}^{3}$ of $0\in\mathbb{R}^{3}$, a submanifold with boundary $D^{\prime}\subset U$ homeomorphic to a closed disc in $\mathbb{R}^{2}$, satisfying $0\in D^{\prime}\subset\{x_{1}=0\}\subset\mathbb{R}^{3}$ and an integral affine isomorphism

  $$(B_{0},\mathscr{A})\cong(U-(D^{\prime}\cup\Delta_{\tau}),\mathscr{A}_{\tau}).$$