Let $B$ be a 3-manifold and let $B_{0}\subseteq B$ be a dense open set such that $\Delta:=B-B_{0}$ is a trivalent graph, i.e. such that $\Delta=\Delta_{d}\cup\Delta_{g}$. Assume that the vertices of $\Delta$ are labeled, i.e. $\Delta_{d}$ decomposes as a union $\Delta_{+}\cup\Delta_{-}$ of positive and negative vertices. Suppose there is a $T^{3}$ bundle $f_{0}:X(B_{0})\rightarrow B_{0}$ such that its local monodromy $\mathcal{M}_{b}$ is generated by

1. 1.

   $T$ as in (2), when $b\in\Delta_{g}$;

2. 2.

   $T_{1},T_{2},T_{3}$ as in (3), when $b\in\Delta_{-}$;

3. 3.

   $(T_{1}^{t})^{-1},(T_{2}^{t})^{-1},(T_{3}^{t})^{-1}$, when $b\in\Delta_{+}$.

Then there is a $T^{3}$ fibration $f:X\rightarrow B$ and a commutative diagram

$$\begin{array}[]{ccc}X(B_{0})&\hookrightarrow&X\\ \downarrow&&\downarrow\\ B_{0}&\hookrightarrow&B.\end{array}$$

Over connected components of $\Delta_{g}$, $(X,f,B)$ is conjugate to the generic singular fibration, over points of $\Delta_{+}$ it is conjugate to the positive fibration and over points of $\Delta_{-}$ to the negative fibration.