K\"{a}hler Soliton Surfaces Are Generically Toric

Let (M, g, ω, f, λ) be a K\"{a}hler gradient Ricci soliton in real dimension four. The first theorem states that it is an integrable Hamiltonian system in a classical sense. Furthermore, it is either of cohomogeneity one or the integrals of motion are given by the potential function f and the scalar curvature S. The second theorem states that it must be toric under a generic assumption. That is, one assumes that the system is non-degenerate and the potential function f is proper. In case λ=0, one further assumes that f is bounded below or above. Then there is an effective, completely integrable Hamiltonian toric T²- action on (M, ω).