arXiv:2610.35932v1 Announce Type: new Abstract: We study the Lane--Emden equation of index $n=5$, the critical polytropic case separating finite-radius from infinite-radius self-gravitating equilibria. The scaling family $\theta_\lambda(\xi)=\lambda^{1/2}\theta(\lambda\xi)$ generates a neutral mode $\phi_0=\tfrac12\theta+\xi\theta'$ about the Schuster profile. Schuster's transformation and a logarithmic variable reduce the equation to the autonomous Hamiltonian system $\ddot\nu=\tfrac14(\nu-\nu^5)$ with a first integral. Besides the Schuster (separatrix) and Srivastava solutions, every orbit crossing $\nu=0$ is given in closed form by a Weierstrass $\wp$-function, and the closed forms agree with direct numerical integration to about $10^{-12}$. Up to scaling and sign, the Schuster solution is the only solution regular at the center, and it separates positive solutions from solutions with infinitely many zeros. In homology-invariant variables the singular solution $\theta=(2\xi)^{-1/2}$ is a center exactly at $n=5$, where its stability changes. For a distance-varying Emden--Fowler system with a shifted equilibrium we prove that at $n=5$ the unique equilibrium is linearly and nonlinearly unstable. Finally, the linearization about the Schuster profile is the Schr\"odinger operator $L=-\Delta_r-5\theta_S^4$ on $L^2((0,\infty),\xi^2d\xi)$. Its Friedrichs extension has essential spectrum $[0,\infty)$, a threshold resonance at $0$ carried by the non-normalizable scaling mode, and exactly one negative eigenvalue, $\lambda_0\approx-1.21$, so the Schuster profile is a saddle point of the associated energy functional.
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