Geometric Characterizations of Generalized $\mathcal{Z}$-solitons on Lorentzian Trans-Sasakian $3$-manifolds

In this paper, we investigate the geometric characterizations of $3$-dimensional Lorentzian trans-Sasakian manifolds admitting a generalized $\mathcal{Z}$-tensor. First, we analyze the behavior of the generalized $\mathcal{Z}$-tensor under certain symmetry conditions, such as Codazzi type, cyclic parallel, and $\phi$-$\mathcal{Z}$ symmetric conditions, and derive the necessary constraints on the manifold structure. We also study the existence of generalized $\mathcal{Z}$-Ricci solitons and prove that the existence of such a soliton implies that the scalar curvature is constant and the soliton generally exhibits a shrinking behavior.