It is well known that the prolongation of an almost complex structure from a manifold $M$ to the tangent bundle of order $r$ on $M$ is also an almost complex structure if it is integrable. The general quadratic structure $F^2=\alpha F+\beta I$ is a generalization of an almost complex structure where $\alpha =0,\beta =-1.$ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $F^2=\alpha F+\beta I,\alpha ,\beta \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. We show that the $r$-lift of the metallic structure $F$ in the tangent bundle of order $r$ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $r$. Moreover, prolongations of $G$-structures immersed in the metallic structure to the tangent bundle of order $r$ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.

In this paper, we investigated $\eta$-Ricci-Bourguignon solitons within the framework of almost pseudo- $W_8$ flat and $M$-projective flat symmetric Lorentzian Kähler space-time manifolds that satisfy the Einstein field equation with and without a cosmological constant. We established necessary and sufficient conditions under which such solitons exhibit expanding, shrinking, or steady behavior. Specifically, we derived constraints on the parameters that govern the soliton dynamics. Furthermore, we extended the results to the case of an $\eta$-Ricci-Bourguignon soliton for dark fluid, dust fluid, stiff matter, and radiational fluid. Our results contribute to the broader understanding of geometric flows and their interaction with the structure of Lorentzian Kähler geometry using partial differential equations.

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo- $W_8$ flat Lorentzian Kähler space-time manifolds.