Spacelike S-Willmore spheres in Lorentzian space forms

We show that spacelike S-Willmore surfaces are the only spacelikeWillmore surfaces with a duality in Lorentzian space forms. We obtain a classification of S-Willmore spheres in Lorentzian conformal space forms. Such a sphere must be congruent to either a complete spacelike stationary (H=0) surface in Rⁿ₁; a super-Willmore sphere in S^2m+2; or a polar transform of a (j-1)-isotropic complete spacelike stationary (H=0) surface in R^2j+2₁, We also show that allWillmore spheres in Q⁴₁ are conformal to a complete spacelike stationary surface in R⁴₁.