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We study some properties of positive solutions to the higher orderconformally invariant equation with a singular set $$ (-\Delta)^m u =u^{\frac{n+2m}{n-2m}}  \textmd{in} ~ \Omega \backslash \Lambda, $$ where$\Omega \subset \mathbb{R}^n$ is an open domain, $\Lambda$ is a closed subsetof $\mathbb{R}^n$, $1 \leq m < n/2$ and $m$ is an integer. We first establishan asymptotic blow up rate estimate for positive solutions near the singularset $\Lambda$ when $\Lambda \subset \Omega$ is a compact set with the upperMinkowski dimension $\overline{\textmd{dim}}_M(\Lambda) < \frac{n-2m}{2}$, oris a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. Wealso show the asymptotic symmetry of singular positive solutions suppose$\Lambda \subset \Omega$ is a smooth $k$-dimensional closed manifold with$k\leq \frac{n-2m}{2}$. Finally, a global symmetry result for solutions isobtained when $\Omega$ is the whole space and $\Lambda$ is a $k$-dimensionalhyperplane with $k\leq \frac{n-2m}{2}$.

Local behavior of positive solutions of higher order conformally invariant equations with a singular set