Baran, Hynek; Marvan, Michal Classification of integrable Weingarten surfaces possessing an \(\mathfrak{sl}(2)\)-valued zero curvature representation. (English) Zbl 1204.53004 Nonlinearity 23, No. 10, 2577-2597 (2010). Summary: We classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an \(\mathfrak{sl}(2)\)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth-century geometers. Finally, we characterize the associated normal congruences. Reviewer: Vladimir I. Oliker (Atlanta) Cited in 4 Documents MSC: 53A05 Surfaces in Euclidean and related spaces 35Q53 KdV equations (Korteweg-de Vries equations) PDFBibTeX XMLCite \textit{H. Baran} and \textit{M. Marvan}, Nonlinearity 23, No. 10, 2577--2597 (2010; Zbl 1204.53004) Full Text: DOI arXiv