  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 $U$.
