= Solution
There is an exact rational-map description of <sigma-model lumps> and a restricted variational rational-map description of <Skyrmions>. The first follows from a <Bogomolny equation>; the second separates angular and radial dependence in a field that generally does not saturate a <Bogomolny bound>.
For the exact example, take the two-dimensional <O3 nonlinear sigma model> with a unit field $\mathbf n$, energy $E=\frac12\int|\nabla\mathbf n|^2d^2x$, and a fixed limit at infinity. Regular fields then have a <one-point compactification> to maps $S^2\to S^2$. Choose the orientation so that the stereographic field
$$
u=\frac{n_1+in_2}{1+n_3},\qquad z=x+iy,
$$
has positive charge when it is <holomorphic>. Equivalently, this is the <CP1 nonlinear sigma model> since the target sphere is the <complex projective line>. Its energy and <topological charge> are
$$
E=4\int\frac{|u_z|^2+|u_{\bar z}|^2}{(1+|u|^2)^2}\,d^2x,
\qquad
N=\frac1\pi\int\frac{|u_z|^2-|u_{\bar z}|^2}{(1+|u|^2)^2}\,d^2x.
$$
Subtracting $4\pi N$ gives $8\int|u_{\bar z}|^2/(1+|u|^2)^2$, so for $N>0$,
$$
\boxed{E\geq4\pi N,\qquad u_{\bar z}=0\text{ at equality}.}
$$
The <Cauchy-Riemann equations> therefore give the minimal-energy fields. A <holomorphic map> from the compactified domain <Riemann sphere> to the target <Riemann sphere> is a <rational map> $u=p(z)/q(z)$, with common <polynomial factors> cancelled. Its <degree of a rational map of the Riemann sphere> is $N=\max(\deg p,\deg q)$. Poles of $u$ are coordinate singularities, not singularities of $\mathbf n$. For instance, $u=(z-a)/\lambda$ with $\lambda\ne0$ is an exact unit lump with energy $4\pi$, arbitrary centre $a$, and scale $|\lambda|$. Its density is $4|\lambda|^2/(|z-a|^2+|\lambda|^2)^2$, whose plane <integral> is $4\pi$. For negative charge use antiholomorphic maps. Scale invariance allows arbitrarily small lumps and does not by itself prevent a singular concentration limit in the time-dependent theory.
The local angular geometry of any degree-$N$ <rational map> is measured by its <angular Jacobian of a rational map>,
$$
J_R(z)=\left[\frac{1+|z|^2}{1+|R|^2}|R'|\right]^2,
\qquad R^*d\Omega=J_Rd\Omega,\qquad\int_{S^2}J_Rd\Omega=4\pi N.
$$
For $R=p/q$, the <Wronskian of a rational map> is
$$
\boxed{W(z)=p'(z)q(z)-p(z)q'(z),\qquad R'=\frac{W}{q^2}.}
$$
At ordinary finite points its zeros mark <ramification points of a holomorphic map>, where the angular density vanishes. At a pole use $q/p$ as the target coordinate, and at infinity use $1/z$ as the domain coordinate. A pole of order $m$ is ramified by $m-1$. The <Riemann-Hurwitz formula> counts total ramification $2N-2$, including infinity, even if the affine <Wronskian> has smaller degree. For $R=z^N$, $W=Nz^{N-1}$ accounts for $N-1$ zeros at zero; the reciprocal coordinate shows the other $N-1$ at infinity.
For the approximate example, the <rational map approximation for Skyrmions> uses spherical radius $r$, angular <stereographic projection> coordinate $z$, and
$$
U(r,z)=\cos f(r)+i\sin f(r)\mathbf n_R(z)\cdot\boldsymbol\sigma,
\qquad
\mathbf n_R=\frac{(2\operatorname{Re}R,2\operatorname{Im}R,1-|R|^2)}{1+|R|^2},
\quad f(0)=\pi,quad f(\infty)=0.
$$
The <Pauli matrices> make this an <SU(2) group>-valued field. The boundary conditions give a continuous centre and the vacuum value at infinity; the profile must also make the energy finite. Angular degree and radial winding factorize to give
$$
\boxed{B=-\frac{2N}{\pi}\int_0^\infty f'\sin^2f\,dr=N.}
$$
At radii with nonzero $-f'\sin^2f$, the angular <baryon number> density is proportional to $J_R$. Thus the <Wronskian of a rational map> locates zero-density directions and reveals the holes or face directions in shell-like Skyrmion configurations.
In dimensionless <Skyrme model> units, insert this ansatz into the quadratic and quartic derivative energies. The angular terms integrate to $4\pi N$ or to the <angular integral in the rational map approximation>,
$$
\mathcal I[R]=\frac1{4\pi}\int_{S^2}J_R^2d\Omega,
\qquad
E=4\pi\int_0^\infty\left[r^2f'^2+2N(f'^2+1)\sin^2f+\mathcal I\frac{\sin^4f}{r^2}\right]dr.
$$
First minimize $\mathcal I$ over degree-$N$ maps, then solve the radial variational equation
$$
(r^2+2N\sin^2f)f''+2rf'+N\sin(2f)(f'^2-1)-\frac{\mathcal I}{r^2}\sin^2f\sin(2f)=0
$$
with the stated boundary conditions. The <Cauchy-Schwarz inequality> gives $\mathcal I\geq N^2$ because $J_R$ has sphere average $N$. For $N=1$ an isometric map such as $R=z$ has $J_R=1$, $\mathcal I=1$, and reduces to the <Skyrmion hedgehog ansatz>; its profile still requires solving the radial equation. For general $N$, angular and radial separation restrict the allowed fields. The minimum within this class is an upper bound on the full sector minimum, not a claim that every exact Skyrmion has this separated form. The original construction is Houghton, Manton and Sutcliffe's https://arxiv.org/abs/hep-th/9705151.
A <rotational symmetry of a rational map> must satisfy $R(gz)=hR(z)$, where $g,h$ are the domain and target rotations written as <Möbius transformations> from <SU(2) matrices>. Since a target rotation is an <isometry>, this identity implies $J_R(gz)=J_R(z)$. For $R=z^N$, $R(e^{i\alpha}z)=e^{iN\alpha}R(z)$, giving axial spatial symmetry accompanied by an internal rotation. This explains the axial symmetry of the degree-two toroidal ansatz.
A useful degree-four example is
$$
R(z)=\frac{z^4+2\sqrt3\,iz^2+1}{z^4-2\sqrt3\,iz^2+1},\qquad
W(z)=8\sqrt3\,iz(1-z^4).
$$
Its finite ramification directions are $0,\pm1,\pm i$, with the sixth at infinity. They are the six coordinate-axis directions under <stereographic projection>, so they are the face normals of a cube. The full map has octahedral rotational equivariance: for example $R(iz)=1/R(z)$, and the cyclic-axis generator $g(z)=(iz+1)/(1-iz)$ obeys $R(gz)=e^{2\pi i/3}R(z)$. These generate the cube's rotational group and give a cubic angular density. The associated profile then produces the familiar cubic charge-four approximation. Symmetry of the <Wronskian> is a useful necessary diagnostic but is not sufficient: replacing this map by $cR$, $c>0$, leaves the same ramification directions, while the identity under $z\mapsto iz$ becomes $cR(iz)=c^2/(cR(z))$, whose target transformation is a sphere rotation only when $c=1$. Equivariance of the entire rational map, rather than symmetry of its critical-point set alone, determines the physical rotational symmetry.
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