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We study qualitative properties for nonnegative solutions to a conformallyinvariant coupled system of fourth order equations involving criticalexponents. For solutions defined in the punctured space, there existessentially two cases to analyze. If the origin is a removable singularity, weprove that non-singular solutions are rotationally invariant and weaklypositive. More precisely, they are the product of a fourth order sphericalsolution by a unit vector with nonnegative coordinates. If the origin is anon-removable singularity, we show that the solutions are radially symmetricand strongly positive. Furthermore, using a Pohozaev-type invariant, we provethe non-existence of semi-singular solutions, that is, all components equallyblow-up in the neighborhood of origin. Namely, they are classified as multiplesof the Emden--Fowler solution. Our results are natural generalizations of thefamous classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm.Pure Appl. Math. (1989)] on the classical singular Yamabe equation.

Qualitative properties for solutions to conformally invariant fourth order critical systems