  We give a new short self-contained proof of the result of Opozda [B. Opozda,
A classification of locally homogeneous connections on 2-dimensional manifolds,
Differential Geom. Appl. 21 (2004), 173-198.] classifying the locally
homogeneous torsion free affine surfaces and the extension to the case of
surfaces with torsion due to Arias-Marco and Kowalski [T. Arias-Marco and O.
Kowalski, Classification of locally homogeneous linear connections with
arbitrary torsion on 2-dimensional manifolds, Monatsh. Math. 153 (2008),
1-18.]. Our approach rests on a direct analysis of the affine Killing equations
and is quite different than the approaches taken previously in the literature.
