Let $(M.g)$ be a Riemannian manifold and $D$ is a distribution on $M$. Assume that $D$ is invariant under the action of the isometry group of $M$.
Under which conditions such $D$ is an integrable distribution?What is an example of a non integrable distribution with this invariant property?
In the case of integrability, under what conditions the leaves of corresponding foliation are totally geodesic?
The motivation comes from the case $\mathbb{R}^{n} \setminus \{0\}$ with the standard metrics. there is only one distribution with this property, which is integrable.