The piecewise smooth Lagrangian fibration $\mathcal{F}=(X,\omega,f,B)$ in Example 5.8 can be perturbed, without changing its topology, so that, for sufficiently big $M$, it becomes smooth on small neighborhoods $N_{h,M}$, $N_{v,M}$ and $N_{d,M}$ of $\Sigma_{h,M}$, $\Sigma_{v,M}$ and $\Sigma_{d,M}$ respectively.