In this paper, we consider classical solutions to the following fractional Liouville system
\(\begin{cases}(-\Delta)^\frac12v_i=\exp{\left(\sum\limits_{j\in I}\gamma^{ij}v_j\right)}\;\;\;\text{in}\;\;\mathbb{R}\\\int_\mathbb{R}\text{e}^vi\;\text{d}s<\infty\end{cases}\;\;\;\text{for all}\;\;i\in\;I,\)
where I = {1,⋯,n}. Assuming that the matrix A = (γij) is nonnegative and invertible, we show that each solution vi is symmetric with respect to the point si. Furthermore, if the matrix A satisfies the irreducibility condition, the points {si} i=1 n coincide. Additionally, given that the matrix A = (γij) satisfies the condition \(\sum\limits_{j\in I}\gamma^{ij}=1\) , along with certain constraints on the masses vector, we employ the method of moving planes to demonstrate that all solutions conform to a standard bubble defined by a common center and scale parameters. And we show the existence of asymmetric solutions to the equations when nonnegativity is absent by applying bifurcation theory.
