Let $\mathcal{F}$ denote the fibration obtained in Lemma 7.4. Given a positive real number $m>M$, there exists a perturbation $\tilde{\mathcal{F}}$ of $\mathcal{F}$ (perhaps defined over a smaller neighborhood of the plane $\{b_{1}=0\}$), such that

• (i)

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

• (ii)

  there are open neighborhoods $B_{h,m}$, $B_{v,m}$ and $B_{d,m}$ of $\Delta_{h,m}$, $\Delta_{v,m}$ and $\Delta_{d,m}$ respectively so that the fibrations $\tilde{\mathcal{F}}_{h,m}$, $\tilde{\mathcal{F}}_{v,m}$ and $\tilde{\mathcal{F}}_{d,m}$ are smooth.