A first-order reduction of the axisymmetric Willmore equation via two first integrals

We derive a first-order formulation of the axisymmetric Willmore equation by combining two independent first integrals, yielding the relation \begin{equation*} \left[\frac{Ψq^2+2q+2C_1ρ}{\sqrt{1-Ψ^2}}\right]^2 +(q^2-2)^2=C_2, \end{equation*} where $C_1$ and $C_2$ are constants of integration. Here $\rho$ denotes the distance from the axis of revolution, and $q=\rho {\mathrm{\Psi }}^{\mathrm{\prime }}-\mathrm{\Psi }$ is the signed hyperbolic curvature of the meridian. We have set $\mathrm{\Psi }=\sin \psi$, where $\psi$ is the tangent angle of the meridian, and a prime denotes differentiation with respect to $\rho$. The above relation is equivalent to a quartic algebraic equation for $\rho {\mathrm{\Psi }}^{\mathrm{\prime }}$ and thus reduces the third-order axisymmetric Willmore equation to a first-order ordinary differential equation on each continuous algebraic branch. This formulation recovers the sphere, axisymmetric minimal surfaces, and the Clifford torus as special cases, while organizing the local solution space according to the pair $(C_1,C_2)$. The same hyperbolic formulation also explains how the spontaneous-curvature term destroys the conservation structure underlying the corresponding first integral for axisymmetric Helfrich surfaces.