In this work, a solitonic study of $LP$-Kenmotsu $n$-manifolds (in brief $(LPK{)}_n$) was performed. Key properties of almost $\ast$-RBSs are established, thus generalizing some of the known results. Additionally, we examine gradient almost $\ast$-RBSs, thereby deriving conditions for the realization of the soliton structure. These results provide new insights into the theory of Ricci-Bourguignon solitons and geometric flows in semi-Riemannian manifolds. It is proven that an $(LPK{)}_n$ that admits an almost $\ast$-RBSs or a gradient almost $\ast$-RBSs is a generalized $\omega$-Einstein spacetime. Moreover, under certain assumptions, an $(LPK{)}_n$ that admits an almost $\ast$-RBSs or a gradient almost $\ast$-RBSs is a perfect fluid spacetime.

In this paper, we consider Egorov and Cahen-Wallach symmetric spaces and study the Riemann solitons on these spaces. We prove that Egorov and Cahen-Wallach symmetric spaces admit the Riemann solitons. Also, we classify the Riemann solitons on these spaces and show that the potential vector fields of the Riemann solitons are Killing, Ricci collineation, and Ricci bi-conformal vector fields.

In the present paper we study Ricci pseudo-symmetry, Z-Ricci pseudo-symmetry and concircularly pseudo-symmetry conditions on a mixed generalized quasi-Einstein spacetime $MG(QE{)}_4$. Also, it is proven that if $d\ne \Lambda$, then $MG(QE{)}_4$ spacetime does not admit heat flux, where $d$ and $\Lambda$ are the function and the cosmological constant, respectively. In the end of this paper we construct a non-trivial example of $MG(QE{)}_4$ to prove its existence.