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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 $\\tilde{g} = f^{-2}g$. 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