An analysis of symmetry groups of generalized $m$-quasi-Einstein manifolds

In this paper, emphasis is placed on how the behavior of the solutions of a system of PDEs is affected by the geometry of generalized $m$-quasi-Einstein manifold, and vice versa. Considering an $n$-dimensional generalized $m$-quasi-Einstein manifold which is conformal to a pseudo-Euclidean space, we find the most general symmetry group of maximal dimension. Moreover, we demonstrate that there is no other low-dimensional invariant on a generalized $m$-quasi-Einstein manifold. As an application, we use the invariant structure of the metric to provide an example of a shrinking $m$-quasi-Einstein manifold.