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Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ canbe realized as a generalized cluster category of quiver with potential. In thisnote, under the assumption that $\mathbb{X}$ has at most three weights or is oftubular 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 abyproduct, a $2$-CY tilted algebra of $\mathcal{C}_\mathbb{X}$ is determined byits 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, wealso obtain a realization of $\mathcal{C}_\mathbb{X}$ viaBuan-Iyama-Reiten-Scott's construction of $2$-CY categories arising frompreprojective algebras.

On cluster categories of weighted projective lines with at most three weights