= Solution
For every real $\alpha$, direct substitution gives the <axially equivariant monomial rational map> identity
$$
\boxed{R(e^{i\alpha}z)=e^{in\alpha}R(z),\qquad R(1/z)=1/R(z).}
$$
The first is a spatial rotation through $\alpha$ about the third axis compensated by a target rotation through $n\alpha$, hence by an <isospin rotation> of the <Skyrmion>. The second pairs the spatial half-turn about the first axis with a target half-turn about that axis. For $n>1$, these generate the continuous axial rotation group together with perpendicular half-turns, often written $D_\infty$; it is the group of <rotations preserving an axis> in the three-dimensional <special orthogonal group>. Under rotations about the axis, the pure spatial stabilizer of the map itself includes the <cyclic group> $C_n$, whereas the larger symmetry just described is combined spatial-target equivariance. Also $R(\overline z)=\overline{R(z)}$ gives a paired reflection symmetry of its angular density.
The <angular Jacobian of a rational map> is
$$
J_R=n^2\frac{|z|^{2n-2}(1+|z|^2)^2}{(1+|z|^{2n})^2}.
$$
It depends only on latitude. Put $u=\log|z|$; then $J_R=n^2\cosh^2u/\cosh^2(nu)$. Since $\cosh(nu)\geq\cosh u$ for $n\geq1$, the maximum is $n^2$ at $u=0$, the equator. For $n>1$ it vanishes at the two poles, producing an axially symmetric ring of angular <Skyrme baryon density>. For $n=1$, $R$ is the identity: every spatial rotation is compensated by the same target rotation, and the <rational map approximation for Skyrmions> becomes the fully <spherically symmetric> <Skyrmion hedgehog ansatz>. \b[The charge and uses are]
$$
\boxed{B=n;\quad n=1\text{ gives the spherical hedgehog},\quad n=2\text{ gives a toroidal two-Skyrmion approximation}.}
$$
The degree-one construction captures the symmetry of the unit <Skyrmion> exactly; its radial profile still has to be solved. The degree-two map supplies the appropriate <toroidal two-Skyrmion> symmetry and useful initial data. Higher monomials give axial charge-$n$ competitors, but axial symmetry need not minimize the energy: the usual lowest-charge examples already include the <tetrahedral three-Skyrmion> and the <cubic four-Skyrmion>. <Topological charge> determines neither shape nor energy optimality by itself.
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