  We provide a complete description of normal affine algebraic varieties over
the real numbers endowed with an effective action of the real circle, that is,
the real form of the complex multiplicative group whose real locus consists of
the unitary circle in the real plane. Our approach builds on the
geometrico-combinatorial description of normal affine varieties with effective
actions of split tori in terms of proper polyhedral divisors on semiprojective
varieties due to Altmann and Hausen

Algebraic Geometry