  In 2008, J. Liberati defined what is a conformal Lie bialgebra and introduced
the conformal classical Yang-Baxter equation (CCYBE). An $L$-invariant solution
to the weak version of CCYBE provides a conformal Lie bialgebra structure. We
describe all solutions to the conformal classical Yang-Baxter equation on the
current Lie conformal algebra $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$ and to
the weak version of it.
