Given $H_{\Delta}\in\mathscr{H}_{\Delta}$, there is a function $H$ defined on a neighborhood of $\Gamma$ whose germ is $H_{\Delta}$, such that for every $\ell\in\mathscr{L}_{\bar{Z}^{\#}_{H}}$, there is a normal form of cylindrical type whose invariants are $(Z^{\#}_{H},\ell,H_{\Delta})$.