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