On the equilibrium of shell membranes under normal loading. Hidden integrability

It is established that a well-known system of classical shell theory descriptive of membranes in equilibrium is, in fact, integrable. The membranes are shown to have geometries within the integrable class of so-called O surfaces. The membrane O surfaces include inter alia minimal, constant mean curvature, constant Gaussian curvature and, more generally, linear Weingarten surfaces, as well as canal surfaces and Dupin cyclides.