Kalnins, E.G.Kress, J.M.Pogosyan, G.Miller, Jr., W.2007-08-162007-08-162000-12https://hdl.handle.net/11299/3510We classify the Hamiltonians H=px2+ py2 +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=J12+J22+ J32+V(x,y,z) on the complex 2-sphere where x2+y2+z2=1. This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.