  We classify hyperbolic polynomials in two real variables that admit a
transitive action on some component of their hyperbolic level sets. Such
surfaces are called special homogeneous surfaces, and they are equipped with a
natural Riemannian metric obtained by restricting the negative Hessian of their
defining polynomial. Independent of the degree of the polynomials, there exist
a finite number of special homogeneous surfaces. They are either flat, or have
constant negative curvature.
