  Let $M$ be a compact complex manifold admitting a K\"ahler structure. A
conformally K\"ahler, Einstein-Maxwell metric (cKEM metric for short) is a
Hermitian metric $\tilde{g}$ on $M$ with constant scalar curvature such that
there is a positive smooth function $f$ with $g = f^2 \tilde{g}$ being a
K\"ahler metric and $f$ being a Killing Hamiltonian potential with respect to
$g$. Fixing a K\"ahler class, we characterize such Killing vector fields whose
Hamiltonian function $f$ with respect to some K\"ahler metric $g$ in the fixed
K\"ahler class gives a cKEM metric $\tilde{g} = f^{-2}g$. The characterization
is described in terms of critical points of certain volume functional. The
conceptual idea is similar to the cases of K\"ahler-Ricci solitons and
Sasaki-Einstein metrics in that the derivative of the volume functional gives
rise to a natural obstruction to the existence of cKEM metrics. However, unlike
the K\"ahler-Ricci soliton case and Sasaki-Einstein case, the functional is
neither convex nor proper in general, and often has more than one critical
points. The last observation matches well with the ambitoric examples studied
earlier by LeBrun and Apostolov-Maschler.
