 Axially equivariant monomial rational map

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.