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][1] 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 $k > 1$. 




  [1]: http://link.springer.com/article/10.1007/s002090050507