  We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber
products of homogeneous Sasakian manifolds. In particular, we prove that SRS
locally induced by a large class of fiber products of homogeneous Sasakian
manifolds are, in fact, $\eta$-Einstein. The results are stronger for
immersions into Sasakian space forms. Moreover, we show an example of a
K\"ahler-Ricci soliton on $\mathbb C^n$ which admits no local holomorphic
isometry into products of homogeneous bounded domains with flat K\"ahler
manifolds and generalized flag manifolds.
