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Indeed, the numerators of Lx (u), Ly (u), and Lz (u) are polynomials in u with real coefficients. Let c be a real (double) zero of L2 (u) and let ax, ay, and az be the remnants from the division of these polynomials by (u-c). Since c is real ax,y,z are also real. It follows from Lx2 (u) + Ly2 (u) + Lz2 (u) = L2 (u) that ax2 + ay2 + az2 =0, i.e., ax = ay = az =0.
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Indeed, the numerators of Lx (u), Ly (u), and Lz (u) are polynomials in u with real coefficients. Let c be a real (double) zero of L2 (u) and let ax, ay, and az be the remnants from the division of these polynomials by (u-c). Since c is real ax,y,z are also real. It follows from Lx2 (u) + Ly2 (u) + Lz2 (u) = L2 (u) that ax2 + ay2 + az2 =0, i.e., ax = ay = az =0.
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Note however that since L2 (u) is conserved by the evolution, the roots of Q2n (u) are constants of motion.
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The frozen separation variables are the solutions of Ls (u) =0, where Ls (u) is given by Eq. 38. Particle-hole symmetry implies Ls (u) =- Ls (-u) and therefore ∑ j=3 n-1 uj =0.
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The frozen separation variables are the solutions of Ls (u) =0, where Ls (u) is given by Eq. 38. Particle-hole symmetry implies Ls (u) =- Ls (-u) and therefore ∑ j=3 n-1 uj =0.
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