First integrals of the axisymmetric shape equation of lipid membranes

The shape equation of lipid membranes [Zhong-can and Helfrich(1987) PRL 59 2486] is a fourth-order partial differential equation. Under the axisymmetric condition, this equation was transformed into a second-order ordinary differential equation (ODE) by Zheng and Liu [Zheng and Liu(1993) PRE 48 2856]. Here we try to further reduce this second-order ODE to a first-order ODE by seeking for first integrals according to the Noether theorem. We indeed find a first integral when the second-order ODE is invariant under conformation transformations. We obtain the mechanical interpretation of the first integral by using the membrane stress tensor.