  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 $(g,\tau)\to (\hat{g},1/\tau)$.
