Homogeneous spaces with geodesic orbit Riemannian metrics and with integrable invariant distributions

We consider two types of homogeneous spaces of Lie groups with compact stabilizer subgroups: those with integrable invariant distributions and those for which all invariant Riemannian metrics are geodesic orbit. Recall that a Riemannian metric is called geodesic orbit if each geodesic is an orbit of a one-parameter subgroup of the full isometry group. We exhibit several homogeneous spaces of the first type that do not belong to the second one.