We describe a general procedure for constructing new Sasaki metrics ofconstant scalar curvature from old ones. Explicitly, we begin with a regularSasaki metric of constant scalar curvature on a 2n+1-dimensional compactmanifold M and construct a sequence, depending on four integer parameters, ofrays of constant scalar curvature (CSC) Sasaki metrics on a compact Sasakimanifold of dimension $2n+3$. We also give examples which show that the CSCrays are often not unique on a fixed strictly pseudoconvex CR manifold or afixed contact manifold. Moreover, it is shown that when the first Chern classof the contact bundle vanishes, there is a two dimensional subcone of SasakiRicci solitons in the Sasaki cone, and a unique Sasaki-Einstein metric in eachof the two dimensional sub cones.
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