Removable singularity of positive solutions for a critical elliptic system with isolated singularity

We study qualitative properties of positive singular solutions to a two-coupled elliptic system with critical exponents. This system is related to coupled nonlinear Schrodinger equations with critical exponents for nonlinear optics and Bose-Einstein condensates. We prove a sharp result on the removability of the same isolated singularity for both two components of the solutions. We also prove the nonexistence of positive solutions with one component bounded near the singularity and the other component unbounded near the singularity. These results will be applied in a subsequent work where the same system in a punctured ball will be studied.