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Abstract In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: {(−Δ)α2u+au=f(v),(−Δ)β2v+bv=g(u),on Ω⫅Rn, $$ \textstyle\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u+au=f(v), \\ (-\Delta)^{\frac{\beta}{2}}v+bv=g(u), \end{cases}\displaystyle \quad \text{on } \Omega\subseteqq \mathbb{R}^{n}, $$ where a,b≥0 $a,b\geq0$. When Ω is the unit ball or Rn $\mathbb{R}^{n}$, we prove that the solutions (u,v) $(u,v)$ are radially symmetric and decreasing. When Ω is the parabolic domain on Rn $\mathbb{R}^{n}$, we prove that the solutions (u,v) $(u,v)$ are increasing. Furthermore, if Ω is the R+n $\mathbb{R}^{n}_{+}$, then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.