explicit constructions of einstein metrics and various generalizations have long been a central problem in differential geometry. i will present a unified description, in the toric framework, for constructing the page einstein metric, the koiso-cao ricci soliton, and the lu-page-pope quasi- einstein metrics on the first del pezzo surface. the existence of quasi-einstein metrics on toric fano manifolds is in general open; our constructions yield some insight into the potential form of such metrics. numerical evidence for the existence of such metrics on the second del pezzo surface will also be outlined, as well as future plans to construct a rigorous proof. this is all joint with s. hall, and partly joint with w. bataat and a. jizany.