This chapter considers the construction of the few known and some new explicit 
solutions to the shape equation. For example, a review of the available 
experimental facts concerning nerve fibers suggests that their form can be 
adequately modelled by Delaunay unduloids. We present parameterizations of these 
surfaces in terms of the geometrical parameters involved in the model. This 
allows for direct expression of all geometric characteristics to the membranes, 
like the length of any separate extension, their volume and surface area via 
explicit formulas. This in turn allows for the direct examination of sensitivity 
of these characteristics on the amplitude oscillations of the parameters and 
animation of the morphological changes. Note that this is either impossible or 
very difficult to achieve by means of numerical analysis. The same applies to 
the modelling of the results in Cole’s experiments, which can be described in 
terms of Delaunay’s nodoids. Besides analytical formulas, we consider the 
geometry of the egg of the sea urchin under bilateral deformation between two 
plates of the model and determine the surface tension of the membrane of the egg 
as a direct consequence of analytical results. Another instance when Delaunay’s 
nodoids appear quite naturally is in the modelling of the membrane fusion. 
According to the commonly-accepted assumption, this process involves an hour-
glass-shaped local contact between two monolayers of opposing membranes, an 
intermediate structure which is called a stalk. The shape of the stalk is 
considered to be an axisymmetrical figure of revolution in the three-dimensional 
space with a planar geometry in the initial configuration. The total energy of 
the stalk is evaluated from the assumption that the stalk is a nodoid with 
constant curvature. The conclusions of the group analysis in the previous 
chapter are used to distinguish two classes of analytical solution of the shape 
equation, the translationally-invariant and axially-symmetric, to be precise. In 
the first case, the Ou-Yang equation is reduced to the equation for a so-called 
generalized elastica, whose solutions are fully described and illustrated with 
graphics. The second class of axially-symmetric membranes is the family of 
Delaunay-like surfaces, but with non-constant mean curvatures. These surfaces 
constitute the first examples of a surfaces with periodic mean curvatures.