  In this paper, we are concerned with a Liouville-type result of the nonlinear
integral equation \begin{equation*}
u(x)=\overrightarrow{l}+C_*\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy.
\end{equation*} Here $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ is a bounded,
uniformly continuous and differentiable function with $k \geq 1$ and
$1<\alpha<n$, $\overrightarrow{l} \in \mathbb{R}^{k}$ is a constant vector, and
$C_*$ is a real constant. If $u$ is the finite energy solution, we prove that
$|\overrightarrow{l}| \in \{0,1\}$. Furthermore, we also give a Liouville type
theorem (i.e., $u \equiv \overrightarrow{l}$).
