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We consider Fano manifolds admitting an algebraic torus action with generalorbit of codimension one. Using a recent result of Datar and Szekelyhidi, weeffectively determine the existence of Kahler-Ricci solitons for thosemanifolds via the notion of equivariant K-stability. This allows us to give newexamples of Kahler-Einstein Fano threefolds, and Fano threefolds admitting anon-trivial Kahler-Ricci soliton.

K-Stability for Fano Manifolds with Torus Action of Complexity One