In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.
Question. Are there non-symmetric homogeneous spaces that share this property?
In this paper the author shows that if we also require that the isometry group act transitively on the set of pairs $(p, \Sigma)$, where $\Sigma$ is a flat and $p$ is a point in it, then the space is symmetric.
My main interest is having many examples, homogeneous or not, of compact Riemannian manifolds for which every geodesic is contained in a totally geodesic, flat torus of dimension $k > 1$.