  Let $(M, g, \omega, f, \lambda)$ 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 $\text{S}$. 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
$\lambda=0$, one further assumes that $f$ is bounded below or above. Then there
is an effective, completely integrable Hamiltonian toric $\mathbb{T}^2$- action
on $(M, \omega)$.
