The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.
"Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons." J. Math. Soc. Japan 70 (1) 25 - 70, January, 2018. https://doi.org/10.2969/jmsj/07017479