On integrability of Weingarten surfaces: a forgotten class

Abstract

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 equation $$z_{yy}+(1/z{)}_{xx}+2=0$$ possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformation to the sine-Gordon equation ${\varphi }_{\xi \eta }=\sin \varphi$. We leave open the problem of finding a Backlund autotransformation and a recursion operator that would produce a local hierarchy.

References in corpus (1)

• On the spectral parameter problem

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