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We classify the rational differential 1-forms with simple poles and simplezeros on the Riemann sphere according to their isotropy group; when the 1-formhas 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-formhas $k\geq 3$ 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 conditionsfor their classification. The classification result enables us to describe themoduli 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 providesufficient (geometrical) conditions for when the 1-forms are isochronous.  Concerning the recent work of J.C.~Langer, we reflect on the strongrelationship between our work and his and provide a partial answer regardingpolyhedral geometries that arise from rational quadratic differentials on theRiemann sphere.

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