  In this paper emphasis is placed on how the behavior of the solutions of a
PDE is affected by the geometry of the generalized $m$-quasi-Einstein manifold,
and vice versa. Considering a $n$-dimensional generalized $m$-quasi-Einstein
manifold which is conformal to a pseudo-Euclidean space, we prove the most
general symmetry group of maximal dimension. Moreover, we demonstrate that
there is no different 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 shrinking $m$-quasi-Einstein manifold
(cf. Example 3). A discussion about the fluid ball conjecture was made.
