It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem was open for a long time.
We use an affine approach to this problem. In [1], it was proved by a direct method that in dimension 2, any homogeneous affine manifold admits a homogeneous geodesic. In [2], it was proved by a differential-topological method that any homogeneous affine manifold admits a homogeneous geodesic through each point. As a corollary, we get the same result for homogeneous pseudo-Riemannian manifolds, either reductive or not.
[1] Dusek, Z., Kowalski, O., Vlasek, Z.: Homogeneous geodesics in homogeneous affine manifolds, Results in Math. (2009).
[2] Dusek, Z.: The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds, J. Geom. Phys. (2010).