More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a possible strategy is that a quadratic vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y)\end{cases}$$ can be rewritten in the form $z'=f(z,\bar{z}) $ with substitution $x=\frac{z+\bar{z}}{2}$ and $y=\frac{z-\bar{z}}{2i}$. He said that this possible strategy does not work for $n>2$. However, he thinks that this strategy leads to finitness of $H(2)$.

My immediate reaction at that time was the following: since hyperbolic geometry concerns the upper half plane, are we implicitly assuming that the upper half plane is invariant under flow? So are we assuming that we have an invariant line? If this is the case then the following fact is an obstruction for continuation:

Fact: every quadratic vector field with an invariant line has at most one limit cycle.

But I think that the story is more complicated. I guess that he was not assuming that the upper half plane is flow invariant. So I guess that there are some thing non trivial in this possible strategy.

I did not understand at all what is his strategy.I frequently asked him for more explanation. But I did not get any answer.

He allowed me to talk with others about his idea.

How does hyperbolic geometry can involve the Hilbert 16th problem? and how does this involvement work only for $n=2$ but not $n>2$.