In this paper, we investigate the geometric structure of Sasakian manifolds that admit smooth solutions to a generalized vacuum static equation (GVSE) involving the contact 1-form. We derive key identities characterizing such solutions and establish several rigidity results. In particular, we show that if the potential function is constant, the manifold becomes \(\eta \) -Einstein. Moreover, for compact connected Sasakian manifolds with constant structure function, the scalar curvature must also be constant. We prove that on an \(\eta \) -Einstein Sasakian manifold, either the scalar curvature vanishes or the potential function remains invariant under the Reeb vector field. Furthermore, we demonstrate that the structure functions defining the \(\eta \) -Einstein condition are necessarily constant throughout the manifold. These results impose strong geometric constraints and highlight the interplay between curvature, potential functions, and contact structures. Finally, an explicit example is constructed on a 5-dimensional Sasakian manifold to illustrate and validate the theoretical framework developed in this work.
