  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.
