  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.

Geometric Topology;Dynamical Systems