We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and To?nnesen-Friedman), arising from a base with a local K?hler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein K?hler metrics (as defined by D. Guan) in all "sufficiently small" admissible K?hler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some K?hler classes while not in others. We also prove an analogous existence theorem for an additional metric type, defined by the requirement that the scalar curvature isan affine combination of a Killing potential and its Laplacian.
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