from $\mathcal I$ and $\mathcal L$ using different pull-back’s; the pre-image by $\phi$ of a point of tangence between $\mathcal I$ and $\mathcal L$ might not be a point of tangency between $\mathcal I^*$ and $\mathcal L^*$. We take the foliations $\mathcal I$ and $\mathcal L$ defined by the equations $LdP-d.PdL=0$ and $(L+ a)dP + d.PdL=0$; their curve of tangencies is defined by $dL\wedge dP=0$ (besides the curve $C_0$). When intersecting with $C_0$, these are the points of tangency of $\mathcal F_0$ with $C_0$.

\[intersection-at-tangencies\] Let $\phi(p)$ be a point of tangency between $\mathcal F_0$ and $C_0$. Then $(tang(\mathcal I^*,\mathcal L^*), C)_p=1$.

We may take local coordinates $(x,y)$ around $p$ and $(r,s)$ around $\phi(p)$ such that

-   $C=\{y=0\}$ and $C_0= \{s=0\}$.

-   $\mathcal F$ and $\mathcal F_0$ are defined by $d(y-x^2)=0$ and $d(s-r^2)=0$ respectively.

The foliations $\mathcal I$ and $\mathcal L$ are defined as $ su\,d(s-r^2)-(s-r^2 + \delta)\,d(su)=0$ and $su\,d(s-r^2)-(s-r^2 + a+\delta)\,d(su)=0$, where $u$ is a holomorphic function such that $u(0,0)\ne 0$ and $\delta \neq 0$. Let us write $\Phi_{F,G}(x,y)= (f(x,y),yA(x,y))$ and $\Phi_{F,H}(x,y)= (g(x,y),yB(x,y))$; we have $A(0,0)\neq 0$, $B(0,0)\ne 0$ and $(f(x,0),0)=(g(x,0),0)= \phi(x)$.

The foliations $\mathcal I^*=\Phi_{F,G}^*{\mathcal I}$ and $\mathcal L^*=\Phi_{F,H}^*{\mathcal L}$ are defined as $$\begin{aligned}
yAu\,d(yA-f^2) - (yA-f^2 +\delta)\,d(yAu)&=&0,\\
