Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equationpossesses a zero curvature representation, a third-order symmetry, and a nonlocal transformation to the sine-Gordon equation φ ξη = sin φ. We leave open the problem of finding a Bäcklund autotransformation and a recursion operator that would produce a local hierarchy.

Abstract. In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an 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.

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Abstract. We present a complete description of 2-dimensional equations that arise as symmetry reductions of fourf 3-dimensional Lax-integrable equations: (1) the universal hierarchy equation uyy = uzuxy − uyuxz; (2) the 3D rdDym equation uty = uxuxy − uyuxx; (3) The basic Veronese web equation uty = utuxy − uyutx; (4) Pavlov's equation uyy = utx + uyuxx − uxuxy.

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