Rotational symmetry of a rational map ($R(gz)=hR(z)$)

Rotations of the domain and target Riemann spheres act as Möbius transformations represented by SU(2) matrices. A combined symmetry requires the equivariance identity $R(gz)=hR(z)$ for the corresponding transformations. Symmetry of the Wronskian of a rational map alone is insufficient: it tests the ramification directions, not the entire map. Target rotations preserve the angular Jacobian of a rational map, so an equivariant map has a domain-invariant angular density.

For positive integer $n$, $R(e^{i\alpha }z)=e^{in\alpha }R(z)$ and $R(1/z)=1/R(z)$. For $n>1$ its angular Jacobian of a rational map vanishes at the poles and has axial symmetry. Its combined rotational group is the group of rotations preserving an axis; for $n=1$ it enlarges to all domain rotations paired with identical target rotations.