  We prove uniqueness of solutions for the nonlocal Liouville equation $$
(-\Delta)^{1/2} w = K e^w \quad \mbox{in $\mathbb{R}$} $$ with finite total
$Q$-curvature $\int_{\mathbb{R}} K e^w \, dx< +\infty$. Here the prescribed
$Q$-curvature function $K=K(|x|) > 0$ is assumed to be a positive,
symmetric-decreasing function satisfying suitable regularity and decay bounds.
In particular, we obtain uniqueness of solutions in the Gaussian case with
$K(x) = \exp(-x^2)$. Our uniqueness proof exploits a connection of the nonlocal
Liouville equation to ground state solitons for Calogero--Moser derivative NLS,
which is a completely integrable PDE recently studied by P. G\'erard and the
second author.
