Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

Abstract: We consider the family $\mathcal{E}(s,r,d)=\Big{ X(z)=\frac{Q(z)}{P(z)}\ e^{E(z)} \frac{\partial}{\partial z} \Big},$ with $Q, P, E$ polynomials, deg${Q}=s$, deg$(P)=r$ and deg$(E)=d$, of singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb{C}}$. For $d\geq1$, $X\in\mathcal{E}(s,r,d)$ has $s$ zeros and $r$ poles on the complex plane and an essential singularity at infinity. Using the pullback action of the affine group $Aut(\mathbb{C})$ and the divisors for $X$, we calculate the isotropy groups $Aut(\mathbb{C}){X}$ and the discrete symmetries for $X\in\mathcal{E}(s,r,d)$. Each subfamily $\mathcal{E}(s,r,d){id}$, of those $X$ with trivial isotropy group in $Aut(\mathbb{C})$, is endowed with a holomorphic trivial principal $Aut(\mathbb{C})$-bundle structure. Necessary and sufficient conditions in order to ensure the equality $\mathcal{E}(s,r,d)=\mathcal{E}(s,r,d)_{id}$ and those $X\in\mathcal{E}(s,r,d)$ with non-trivial isotropy are realized. Explicit global normal forms for $X\in\mathcal{E}(s,r,d)$ are presented. A natural dictionary between vector fields, 1-forms, quadratic differentials and functions is extended to include the presence of non-trivial discrete symmetries $\Gamma<Aut(\mathbb{C})$.