  For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation
\[ \Delta^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\}
\] with a non-removable singularity at the origin. We show that
$|x|^{\frac{n-4}{2}} u$ is a periodic function of $\ln |x|$ and we classify all
periodic functions obtained in this way. This result is relevant for the
description of the asymptotic behavior near singularities and for the
$Q$-curvature problem in conformal geometry.
