Uniqueness and Symmetry Breaking for Singular $Q$-curvature Equations in $\R^n$

For $n\ge3$ and $γ>-1$, we study normal solutions of \begin{equation*} (-Δ)^{n/2}u=(n-1)!\,|x|^{nγ}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove exist