Special K\"ahler-Ricci potentials on compact K\"ahler manifolds

A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant $C^\infty$ function $\tau$ such that $J(\nabla\tau)$ is a Killing vector field and, at every point with $d\tau\ne 0$, all nonzero tangent vectors orthogonal to $\nabla\tau$ and $J(\nabla\tau)$ are eigenvectors of both $\nabla d\tau$ and the Ricci tensor. For instance, this is always the case if $\tau$ is a nonconstant $C^\infty$ function on a K\"ahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde g=g/\tau^2$, defined wherever $\tau\ne 0$, is Einstein. (When such $\tau$ exists, $(M,g)$ may be called {\it almost-everywhere conformally Einstein}.) We provide a complete classification of compact K\"ahler manifolds with special K\"ahler-Ricci potentials and use it to prove a structure theorem for compact K\"ahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.