Let $\mathcal{F}$ be the fibration obtained in Lemma 7.6. There is a perturbation $\tilde{\mathcal{F}}$ of $\mathcal{F}$ such that:

• (i)

  $\tilde{\mathcal{F}}$ is topologically conjugate to $\mathcal{F}$;

• (ii)

  there exists a submanifold with boundary $D\subset B$, homeomorphic to a closed disc in $\mathbb{R}^{2}$, with $\Delta\cap(B-D)$ consisting of three disjoint segments, such that $\tilde{\mathcal{F}}|_{\mathbb{R}^{3}-D}$ is a smooth Lagrangian fibration.