All known examples of gradient Kähler-Ricci soliton in real dimension four are toric and the symmetry is intrinsically related to the potential function f and the scalar curvature \({\mathrm S}\) . In this article, we consider the case that f and \({\mathrm S}\) are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.
