In this paper, the authors studied qualitative properties of positive singular solutions to the following two-coupled elliptic system with critical exponents \[ \begin{cases} -\Delta u=\mu_1u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, &\quad x\in\mathbb R^N\backslash\{0\},\\ -\Delta v=\mu_2v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1}u^{\frac{2^\ast}{2}},&\quad x\in\mathbb R^N\backslash\{0\},\\ u,v>0\mathrm{ and }u,v\in C^2(\mathbb R^N\backslash\{0\}).\end{cases} \] Here, \(\mu_1\), \(\mu_2\) and \(\beta\) are all positive constants, \(N\geq 3\) and \(2^\ast:=\frac{2N}{N-2}\) is the Sobolev critical exponent. They generalize a classical result of removable singularity to the system above. Moreover, they also prove the nonexistence of positive solutions with one component bounded near 0 and the other component unbounded near 0.