Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms

Let (g, X) be a Sasaki-Ricci soliton on a Sasakian manifold S. We prove that if (S, g) admits a local Sasakian immersion in a Sasakian space form S(N, c) of constant phi-sectional curvature c, then S is eta-Einstein and its eta-Einstein constants are a linear function of c with rational coefficients. Moreover, if c <= -3, S is locally equivalent to the Sasakian space form S(n, c). Further results are obtained in the compact setting, i.e. when c > -3, under additional hypotheses.