Rigidity theorems for generic holomorphic germs of dicritic foliations and vector fields in $(\mathbb C^2,0)$

We consider the class $\mathcal V_{n+1}^d$ of dicritical germs of holomorphic vector fields in $(\mathbb C^2,0)$ with vanishing $n$-jet at the origin for $n\ge 1$. We prove, under some genericity assumptions, that the formal equivalence of two generic germs implies their analytic equivalence. A similar result is also established for orbital equivalence. Moreover, we give formal, orbitally formal, and orbitally analytic classifications of generic germs in $\mathcal V_{n+1}^d$ up to a change of coordinates with identity linear part.