  Let $\mathbb{X}$ be a weighted projective line and
$\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We
classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such
that $\tau^2 T\cong T$, where $\tau$ is the Auslander-Reiten translation in
$D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we
classify the 2-representation-finite algebras which are derived-equivalent to a
canonical algebra. This complements Iyama-Oppermann's classification of the
iterated tilted 2-representation-finite algebras. By passing to 3-preprojective
algebras, we obtain a classification of the selfinjective cluster-tilted
algebras of canonical-type. This complements Ringel's classification of the
selfinjective cluster-tilted algebras.
