  k-Curvature homogeneous three-dimensional Walker metrics are described for
k=0,1,2. This allows a complete description of locally homogeneous
three-dimensional Walker metrics, showing that there exist exactly three
isometry classes of such manifolds. As an application one obtains a complete
description of all locally homogeneous Lorentzian manifolds with recurrent
curvature. Moreover, potential functions are constructed in all the locally
homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.
