  We examine the space of solutions to the affine quasi--Einstein equation in
the context of homogeneous surfaces. As these spaces can be used to create
gradient Yamabe solitions, conformally Einstein metrics, and warped product
Einstein manifolds using the modified Riemannian extension, we provide very
explicit descriptions of these solution spaces. We use the dimension of the
space of affine Killing vector fields to structure our discussion as this
provides a convenient organizational framework.
