$\eta$-Ricci-Bourguignon solitons in an almost pseudo- $W_8$ flat and $M$-projective flat symmetric Lorentzian Kähler space-time manifold

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.