On a manifold of dimension at least six, let $(g,��)$ be a pair consisting of a K��hler metric g which is locally K��hler irreducible, and a nonconstant smooth function $��$. Off the zero set of $��$, if the metric $\hat{g}=g/��^2$ is a gradient Ricci soliton which has soliton function $1/��$, we show that $\hat{g}$ is K��hler with respect to another complex structure, and locally of a type first described by Koiso. Moreover, $��$ is a special K��hler-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 $(g,��)$ satisfying a Ricci-Hessian equation is invariant, in a suitable sense, under the map $(g,��)\to (\hat{g},1/��)$.

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