Suppose we are given two normal forms of cylindrical type $\mathcal{F}_{u,H}$ and $\mathcal{F}_{u^{\prime},H^{\prime}}$. From the results in [1] (cf. also Theorem 4.13), a necessary condition for $f_{H}$ and $f_{H^{\prime}}$ to be symplectically conjugate is that $H_{\Delta}=H^{\prime}_{\Delta}$, so suppose this holds. This gives a symplectomorphism, which we denote by $\Phi_{H,H^{\prime}}$, between the total spaces $X$ and $X^{\prime}$ of the two fibrations which conjugates $(X,f_{H},B)$ and $(X^{\prime},f_{H^{\prime}},B^{\prime})$. By pulling back $(V^{\prime},u^{\prime})$ via this symplectomorphism and computing the Taylor series, we obtain a sequence of fibrewise closed sections of $\mathfrak{L}^{\ast}$ which we call $\Phi_{H,H^{\prime}}\cdot\ell^{\prime}$. Using the same arguments as in the proof of Theorem 6.12 (cf.[2], Theorem 6.11), we can then show that $\mathcal{F}_{u,H}$ and $\mathcal{F}_{u^{\prime},H^{\prime}}$ are symplectically conjugate if and only if $\Phi_{H,H^{\prime}}\cdot\ell^{\prime}=\ell$. In particular, when $H=H^{\prime}$, they are symplectically conjugate if and only if $\ell=\ell^{\prime}$.