On integrability of Weingarten surfaces: a forgotten class

TL;DR

This paper rediscovered a forgotten class of integrable Weingarten surfaces, demonstrating their properties and disproving a longstanding conjecture, while highlighting open problems in their integrability structure.

Contribution

It identifies a previously overlooked class of integrable surfaces and analyzes their associated nonlinear PDE, providing new insights into their integrability and structure.

Findings

Disproved the Finkel-Wu conjecture.

Established zero curvature representation for the PDE.

Connected the PDE to the sine-Gordon equation.

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 $\phi_{\xi\eta} = \sin\phi$. We leave open the problem of finding a Backlund autotransformation and a recursion operator that would produce a local hierarchy.