Generalizations of Kähler-Ricci solitons on projective bundles

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman), arising from a base with a local Kahler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein Kahler metrics (as defined by D. Guan) in all "sufficiently small" admissible Kahler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some Kahler 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.