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We obtain a structure theorem for the group of holomorphic automorphisms of a conformally Kähler, Einstein–Maxwell metric, extending the classical results of Matsushima (in: Conference board of the mathematical sciences regional, conference series in mathematics, no. 7, American Mathematical Society, Providence, 1971), Licherowicz (Géométrie des groupes de transformation, Dunod, 1958), and Calabi (Extremal...