Geometric structures over non-reductive homogeneous 4-spaces

We investigate the geometric properties of four-dimensional non-reductive pseudo-Riemannian manifolds admitting an invariant metric of signature $(2,2)$. In particular, we obtain the classification of Walker structures, self-dual and anti-self-dual metrics and para-Hermitian structures for all the invariant metrics of these manifolds. For the examples admitting an invariant parallel null plane distribution, we shall also obtain an explicit description of the invariant metrics in canonical Walker coordinates.