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We study the asymptotic behavior for singular solutions to a critical fourthorder system generalizing the constant $Q$-curvature equation. Our main resultextends to the case of strongly coupled systems, the celebrated asymptoticclassification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. PureAppl. Math. (1989)] and [N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen,Invent. Math., (1999)]. On the technical level, we use an involved spectralanalysis to study the Jacobi fields' growth properties in the kernel of thelinearization of our system around a blow-up limit solution. Besides, we obtainsharp a priori estimates for the decay rate of singular solutions near theorigin. Consequently, we prove that sufficiently close to the isolatedsingularity solutions behave like the so-called Emden--Fowler solution. Ourmain theorem positively answers a question posed by [R. L. Frank and T.K\"onig, Anal. PDE (2019)] concerning the local behavior close to the isolatedsingularity for scalar solutions in the punctured ball.

Asymptotics for singular solutions to conformally invariant fourth order systems in the punctured ball