A free action of $\mathbb{Z}/3\mathbb{Z}$ on a Riemannian manifold $(M, g)$ is called an equilateral action if for every $x\in M$ all three points of orbit of $x$ have the same distance from each other.

What is an example of a Riemannian manifold which does not admit such an action but it already admit a smooth free action by cyclic group of order $3$?