We take as coordinates $(b_{1},b_{2},b_{3})$ on $B$ the ones given by the normalization of the singularity in Theorem 4.6. Then the proof goes essentially as in Proposition 4.8. As in the smooth case, one can define $\gamma$ as being represented by an $3$-tuple of sections $b\mapsto(\gamma_{1}(b),\gamma_{2}(b),\gamma_{3}(b))$, each one given by certain composition of Hamiltonian flows. In this case, however, $b\mapsto\gamma_{2}(b)$ does not vary smoothly but piecewise smoothly, failing to be smooth along $\Gamma$. The contribution of the path $\gamma_{2}\cap\mathfrak{U}$ to the periods $\lambda_{2}^{\pm}$ is $\lambda_{0}$. On the other hand, the contribution of $\gamma_{2}\cap X-\mathfrak{U}$ is $dH^{\pm}$. In contrast, the other two periods can be computed along paths entirely contained in $\mathfrak{U}$ which implies that they are smoothly defined on $B$. ∎