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

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary invariant Riemannian metric on the space, every geodesic is an orbit of a 1-parameter subgroup of the isometry group. We found several homogeneous spaces of the first type that are not spaces of the second type. Among them there are several homogeneous spaces that admit invariant Einstein metrics.

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