Completeness of superintegrability in two-dimensional constant curvature spaces

We classify the Hamiltonians \(H=p_x^2+p_y^2+V(x,y)\) of all classical superintegrable systems in two dimensional complex Euclidean space with second-order constants of the motion. We similarly classify the superintegrable Hamiltonians \(H=J_1^2+J_2^2+J_3^2+V(x,y,z)\) on the complex 2-sphere where \(x^2+y^2+z^2=1\). This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.