map). We start with a line bundle of Chern class $n\in \mathbb N$ over a curve $C$; the lines of the bundle define a foliation $\mathcal L$ in the total space of the bundle. Let $f: C \rightarrow \Pp^1$ be a ramified map with simple critical points and let $p$ be one of these points. There exists an involution $i$ defined in a neighborhood of $p$ in $C$ by $f(q)= f(i(q))$ for $q$ close to $p$.

We fix a neighborhood $U$ of $p$ and a holomorphic diffeomorphism $H:U \rightarrow {\mathbb D}\times {\mathbb D}$ such that: 1) $H(p)= (0,0)$; 2) $H(C \cap U)=\{(z_1,0)\in {\mathbb D}\times {\mathbb D}\}$; 3)$H$ takes $\mathcal L$ to the foliation $dz_1=0$ and 4) $h:= H|_{C \cap U}$ conjugates $i$ to the involution $z \longmapsto -z$, that is: $h(i(q))=-h(q)$. We take also a biholomorphism $\psi$ from ${\mathbb D}\times {\mathbb D}$ to a neighborhood of $(0,0)$ with the properties: 1) $\psi (0,0)=(0,0)$; 2) $\psi(z_1,0)= (z_1,0)$; 3) $\psi(\{1/2<|z_1|<1\}\times {\mathbb D})$ is saturated by leaves of the foliation $dZ_2-Z_1dZ_1=0$ and 4) $\psi$ is a holomorphic diffeomorphism when restricted to $\{1/2 < |z_1| <1\}\times {\mathbb D}$ that sends the foliation $dz_1=0$ to the foliation $dZ_2-Z_1dZ_1=0$. Put $H_1= \psi \circ H$.

We remove 