Dynamics of singular complex analytic vector fields with essential singularities II

The singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb C}_z$ belonging to the family ${\mathscr E}(r,d)=\left\{ X(z)=\frac{1}{P(z)} e^{E(z)}\frac{\partial }{\partial z}\ \Big\vert \ P, E\in\mathbb{C}[z]\right\}$, where $P$ is monic, $deg(P)=r$, $deg(E)=d$, $r+d\geq 1$, have a finite number of poles on the complex plane and an isolated essential singularity at infinity (for $d\geq 1$). Our aim is to describe geometrically $X$, particularly the singularity at infinity. We use the natural one to one correspondence between $X$, a global singular analytic distinguished parameter $\Psi_X(z)=\int^z P(\zeta) e^{-E(\zeta)}d\zeta$, and the Riemann surface ${\mathcal R}_X$ of this distinguished parameter. We introduce $(r,d)$-configuration trees which are weighted directed rooted trees. An $(r,d)$-configuration tree completely encodes the Riemann surface ${\mathcal R}_X$ and the singular flat metric associated on ${\mathcal R}_X$. The $(r,d)$-configuration trees provide "parameters" for the complex manifold ${\mathscr E}(r,d)$, which give explicit geometrical and dynamical information; a valuable tool for the analytic description of $X\in{\mathscr E}(r,d)$. Furthermore, given $X$, the phase portrait of the associated real vector field $Re(X)$ on the Riemann sphere is decomposed into $Re(X)$-invariant components: half planes and finite height strips. The germ of $X$ at infinity is described as a combinatorial word (consisting of hyperbolic, elliptic, parabolic and entire angular sectors having the point at infinity of $\widehat{\mathbb C}_z$ as center). The structural stability, under perturbation in ${\mathscr E}(r,d)$, of the phase portrait of $Re(X)$ is characterized by using the $(r,d)$-configuration trees. We provide explicit conditions, in terms of $r$ and $d$, as to when the number of topologically equivalent phase portraits of $Re(X)$ is unbounded.