  Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$
the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can
be realized as a generalized cluster category of quiver with potential. In this
note, under the assumption that $\mathbb{X}$ has at most three weights or is of
tubular type, we prove that if the generalized cluster category
$\mathcal{C}_{(Q,W)}$ of a Jacobi-finite non-degenerate quiver with potential
$(Q,W)$ shares a $2$-CY tilted algebra with $\mathcal{C}_\mathbb{X}$, then
$\mathcal{C}_{(Q,W)}$ is triangle equivalent to $\mathcal{C}_\mathbb{X}$. As a
byproduct, a $2$-CY tilted algebra of $\mathcal{C}_\mathbb{X}$ is determined by
its quiver provided that $\mathbb{X}$ has at most three weights. To this end,
for any weighted projective line $\mathbb{X}$ with at most three weights, we
also obtain a realization of $\mathcal{C}_\mathbb{X}$ via
Buan-Iyama-Reiten-Scott's construction of $2$-CY categories arising from
preprojective algebras.
