Kahler Gradient Ricci Solitons (KGRS) are fundamental in the theory of Ricci flows. This talk's focus is on complex dimension two and will first review the recent beautiful classification of the shrinking possibly non-compact case. Then we'll discuss an approach to detect symmetry applicable to all cases. This differs from and should complement the popular perspective based on asymptotic behaviors. The idea is inspired by Morse-theoretic aspects of symplectic geometry and involves the understanding of singular sets of a moment map. Precisely, we'll show that a KGRS in complex dimension two is an integrable Hamiltonian system; if the system is generic, then it admits a holomorphic 2-torus action.