Generalized Quasi-Einstein metrics on admissible manifolds

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and T{\o}nnesen-Friedman), arising from a base with a local K"ahler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein K"ahler metrics (as defined by D. Guan) in all “sufficiently small” admissible K"ahler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some K"ahler 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 is an affine combination of a Killing potential and its Laplacian.