Reductions of Gauss-Codazzi equations

We prove that conformally parametrized surfaces in Euclidean space $\text{Rcubec}$ of curvature $c$ admit a symmetry reduction of their Gauss-Codazzi equations whose general solution is expressed with the sixth Painlev'e function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko, Eitner and Kitaev 1997) can be recovered by such a reduction.