The isotropy group of a foliation: the local case

Given a holomorphic singular foliation $\text{fa}$ of $({\text{C}}^n,0)$ we define $Iso(\text{fa})$ as the group of germs of biholomorphisms on $({\text{C}}^n,0)$ preserving $\text{fa}$: $Iso(\text{fa})=\{\mathrm{\Phi }\in Diff({\text{C}}^n,0)\mathrm{\mid }{\mathrm{\Phi }}^{\ast }(\text{fa})=\text{fa}\}$. The normal subgroup of $Iso(\text{fa})$, of biholomorphisms sending each leaf of $\text{fa}$ into itself, will be denoted as $Fix(\text{fa})$. The corresponding groups of formal biholomorphisms will be denoted as $\text{wh}Iso(\text{fa})$ and $\text{wh}Fix(\text{fa})$, respectively. The purpose of this paper will be to study the quotients $Iso(\text{fa})\mathrm{/}Fix(\text{fa})$ and $\text{wh}Fix(\text{fa})\mathrm{/}\text{wh}Fix(\text{fa})$, mainly in the case of codimension one foliation.