  In this paper, we classify the compact locally homogeneous non-gradient
$m$-quasi Einstein 3-manifolds. Along the way, we prove that given a compact
quotient of a Lie group of any dimension that is $m$-quasi Einstein, the
potential vector field $X$ must be left invariant and Killing. We also classify
the nontrivial $m$-quasi Einstein metrics that are a compact quotient of be the
product of two Einstein metrics. We also show that $S^1$ is the only compact
manifold of any dimension which admits a metric which is nontrivially $m$-quasi
Einstein and Einstein.
