Spheres serve in different geometries as the most symmetric models. In CR-geometry a sphere is given by the equation Im $w= |z|^2$ (which is essentially equivalent to $|z|^2+|w|^2=1$). In this geometry the sphere has an 8-dimensional family of fractional-linear transformations of the ambient space. Choosing one symmetry out of a 7-dimensional subfamily gives the sphere the additional structure of a Sasaki manifold with a canonical embedding into $C^2$.
In my talk I will answer the question, posed by N. Stanton, how many such embeddings exist. This is joint work with V. Ezhov.