The talk is based on a joint work with Nataliya Goncharuk.

Consider the set of vector fields in $\mathbb{C}^2$ given by \[ \dot x = P(x, y),\quad \dot y = Q(x, y), \] where $P$ and $Q$ are polynomials of degree at most $n$. Such vector field generates a singular foliation of $\mathbb{C}^2$. The leaves of this foliation have complex dimension one, and real dimension two.

It turns out that the properties of a *generic* foliations of this type are very different from the properties of a generic polynomial foliation of $\mathbb{R}^2$.

In particular, for a generic foliation:
1) All its leaves are dense in $\mathbb{C}^2$.
2) It has an infinite number of independent complex limit cycles (i.e. loops on the leaves with non-identical holonomy maps); in particular, some leaves are not topological discs.
3) It is rigid.

We shall start with a survey of classical results, then describe our contribution to 2) and to the question about the genera of the leaves.