Rota-Baxter operators on $\mathrm{C}\mathrm{u}\mathrm{r}({\mathrm{s}\mathrm{l}}_2(\mathbb{C}))$

We classify all Rota-Baxter operators on the simple conformal Lie algebra $\mathrm{C}\mathrm{u}\mathrm{r}({\mathrm{s}\mathrm{l}}_2(\mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang-Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.