Special Ricci-Hessian equations on Kähler manifolds

Special Ricci-Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367–380] involve functions $\tau$ on $M$ and state that, for some function $\alpha$ of the real variable $\tau$, the sum of $\alpha \mathrm{\nabla }d\tau$ and the Ricci tensor equals a functional multiple of the metric $g$, while $\alpha \mathrm{\nabla }d\tau$ itself is assumed to be nonzero almost everywhere. Three well-known obvious standard'' cases are provided by (non-Einstein) gradient Kähler-Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler-Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $α=2\cotτ$, or $α=2\cothτ$, or $α=2\tanhτ$. We also use the Cartan-Kähler theorem to prove that these three types are actually realized in a nonstandard’’ way.