Classification of rational differential forms on the Riemann sphere, via their isotropy group

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$ poles the isotropy group is finite. In particular we show that all the finite subgroups of $PSL(2,\mathbb{C})$ are realizable as isotropy groups for a rational 1-form on $\widehat{\mathbb{C}}$. We also present local and global geometrical conditions for their classification. The classification result enables us to describe the moduli space of rational 1-forms with finite isotropy that have exactly $k$ simple poles and $k-2$ simple zeros on the Riemann sphere. Moreover, we provide sufficient (geometrical) conditions for when the 1-forms are isochronous. Concerning the recent work of J.C.~Langer, we reflect on the strong relationship between our work and his and provide a partial answer regarding polyhedral geometries that arise from rational quadratic differentials on the Riemann sphere.