A purely analytic derivation of Bonnet surfaces

Abstract

Bonnet has characterized his surfaces by a geometric condition. What is done here is a characterization of the same surfaces by two analytic conditions: (i) the mean curvature $H$ of a surface in ${\mathbb{R}}^3$ should admit a reduction to an ordinary differential equation; (ii) this latter equation should possess the Painlevé property.

7 pages, to appear, Journal of geometry and physics