  We classify all regular three-dimensional convex cones which possess an
automorphism group of dimension at least two, and provide analytic expressions
for the complete hyperbolic affine spheres which are asymptotic to the
boundaries of these cones. The affine spheres are represented by explicit
hypersurface immersions into three-dimensional real space. The generic member
of the family of immersions is given by elliptic integrals.
