  We show that if a Fano manifold $M$ is K-stable with respect to special
degenerations equivariant under a compact group of automorphisms, then $M$
admits a K\"ahler-Einstein metric. This is a strengthening of the solution of
the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and
can be used to obtain new examples of K\"ahler-Einstein manifolds. We also give
analogous results for twisted K\"ahler-Einstein metrics and Kahler-Ricci
solitons.
