The energy of the bending of the membranes within the Canham model is directly 
justified as an extension of the case of bending of the elastic strips known as 
Euler’s Elasticas. Later, this model was elaborated upon by Helfrich & Deuling 
in the form of a non-linear system of two equations for the curvatures of the 
axially-symmetric membranes. Finally, Ou-Yang and Helfrich introduced the model 
which is currently seen to be most adequate for the description of membrane 
configurations. Since we are primarily interested in analytical solutions, it is 
quite natural that the equation by Ou-Yang and Helfrich be examined for the 
presence of symmetries, because they are in direct connection with solutions. 
When this is done, it becomes clear that the most general group of symmetries of 
the shape equation of the form coincides with the group of Euclidean motions in 
the real three-dimensional space. Among the generators of this group are those 
of rotations and translations, which hint about the existence of the analytical 
solutions discussed in the present and the subsequent chapters.