Nondegeneracy of half-harmonic maps from $\mathbb{R}$ into $\mathbb{S}^1$

We prove that the standard half-harmonic map $U:\mathbb{R}\to\mathbb{S}^1$ defined by \begin{equation*} x\to \begin{pmatrix} \frac{x^2-1}{x^2+1} \frac{-2x}{x^2+1} \end{pmatrix} \end{equation*} is nondegenerate in the sense that all bounded solutions of the linearized half-harmonic map equation are linear combinations of three functions corresponding to rigid motions (dilation, translation and rotation) of $U$.