  We construct multiple solutions to the nonlocal Liouville equation
\begin{equation} \label{eqk} \tag{L} (-\Delta)^{\frac{1}{2}} u = K(x) e^u \quad
\mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form
$K(x) = 1+\varepsilon \kappa(x)$ with $\varepsilon \in (0,1)$ small and $\kappa
\in C^{1,\alpha}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $\alpha >
0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from
the bubbles. These solutions provide examples of flat metrics in the half-plane
with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they
imply the existence of multiple ground state soliton solutions for the
Calogero-Moser derivative NLS.
