Special K\"ahler-Ricci potentials and Ricci solitons

On a manifold of dimension at least six, let $(g,\tau)$ be a pair consisting of a K\"ahler metric g which is locally K\"ahler irreducible, and a nonconstant smooth function $\tau$. Off the zero set of $\tau$, if the metric $\hat{g}=g/\tau^2$ is a gradient Ricci soliton which has soliton function $1/\tau$, we show that $\hat{g}$ is K\"ahler with respect to another complex structure, and locally of a type first described by Koiso. Moreover, $\tau$ is a special K\"ahler-Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci-Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs $(g,\tau)$ satisfying a Ricci-Hessian equation is invariant, in a suitable sense, under the map $(g,\tau)\to (\hat{g},1/\tau)$.