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 $\widehat{g}=g\mathrm{/}{\tau }^2$ is a gradient Ricci soliton which has soliton function $1\mathrm{/}\tau$, we show that $\widehat{g}$ 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 $(g,\tau )\to (\widehat{g},1\mathrm{/}\tau )$.