  Special Ricci-Hessian equations on K\"ahler 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 $\alpha\nabla d\tau$ and the Ricci tensor equals a functional
multiple of the metric $g$, while $\alpha\nabla d\tau$ itself is assumed to be
nonzero almost everywhere. Three well-known obvious ``standard'' cases are
provided by (non-Einstein) gradient K\"ahler-Ricci solitons,
conformally-Einstein K\"ahler metrics, and special K\"ahler-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, $\alpha=2\cot\tau$, or
$\alpha=2\coth\tau$, or $\alpha=2\tanh\tau$. We also use the Cartan-K\"ahler
theorem to prove that these three types are actually realized in a
``nonstandard'' way.
