  We extend to the Sasakian setting a result of Tian and Zhu about the
decomposition of the Lie algebra of holomorphic vector fields on a K\"ahler
manifold in the presence of a K\"ahler-Ricci soliton. Furthermore we apply
known deformations of Sasakian structures to a Sasaki-Ricci soliton to obtain a
stability result concerning \emph{generalized} Sasaki-Ricci solitons,
generalizing Li in the K\"ahler setting and also He and Song by relaxing some
of their assumptions.
