Elastic curves and Willmore Hopf-Tori

Polynomial Killing fields and spectral curvePolynomial Killing fieldsPolynomial Killing fieldsA polynomial Killing field of a curve on S 2 is a polynomialX =d∑X k λ k where X d =k=0( )i 00 −iwith X k : R → su 2 smooth that satisfies a Lax equationX ′ = 1 2 [X, V (X)] with V (X) := X d−1 + X d λ ,and is twisted (or anti-twisted), so thatX(−λ) = ± ( ) (0 1−1 0 X(λ) 0 1)−1 0 .