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Relation Between Shape Equations for Axisymmetric vesicles

In deriving the shape equation for axisymmetric vesicles from the Helfrich free energy the variation must be taken with respect to the contour. It is pointed out that the widely used Helfrich shape equation and the other one given by Peterson were derived by improper variations, hence both equations need to be examined. The validity of the two equations is discussed based on analysis of their relation revealed by two derived first integrals. A newly found exact solution for the general shape equation is studied and the relation of this exact solution of spherical topology with that of the Helfrich shape equation is discussed.