Let $V=\sum\_{i=1}^{n}a\_i(z\_1,\ldots z\_n)\frac{\partial}{\partial z\_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of $a\_i$ are the origin, i.e. $V$ has isolated singularity. We know that if $V$ is a radial vector field $\sum\_{i=1}^{n}z\_i\frac{\partial}{\partial z\_i}$, then there are infinite separatrix through the origin. My question is: is this the only case that admit infinite separatrix? If there are other types of vector fields which have infinite separatrix, what is the condition?



Holomorphic vector field with infinite separatrix
No, a lot more vector fields have infinitely many separatrices. They are called «dicritical». For instance, you can change locally the analytic coordinates $(z\_1,\ldots,z\_n)$, the vector field will not be radial anymore but still it will admit infinitely many separatrices (all oft its trajectories, actually). Aside this trivial case, you can determine if the vector field is dicritical by performing a (succesive chain of) blow-up of the singularity. This is an algebraic process and the dicriticality condition amounts to the vanishing of some term in the power series expansion in the new coordinates. This is classical stuff for $n=2$ (Seidenberg algorithm) and tending-to-classical for $n=3$ (Cano, Panazzolo-MacQuillan). For $n>3$ not much is known I think.


