Consider a torus $T^n$. An differential operator $\mathcal{O}$ acts on differential forms on $T^n$. Let $R$ be a smooth vector field on $T^n$, whose orbits are dense on $T^n$ (for instance, irrational flow on $T^2$).

My question is: if $\mathcal{L}_R$ and $ \mathcal{O}$ commutes, does it imply $\mathcal{O}$ also commutes with all $n$ generators of torus action (namely $\partial_{\varphi_i}$ in coordinate)?

I tested for example $n = 2$, and $\mathcal{O} = \mathcal{L}_X$ for some smooth vector field $X$, and $R = p\partial_{\varphi_1} + q \partial_{\varphi_2}$ where $p$ and $q$ are co-irrational. Then $\mathcal{L}_R\mathcal{O}=\mathcal{O}\mathcal{L}_R$ implies $X$ can only has constant components and therefore must commute with $\partial_{\varphi_i}$. But I don't know how to generalize.

Thank you!

=================

**Edit**: as pointed out by Sergei Ivanov, the answer to original question is "No". Now let me put in a bit more assumptions. Put a standard flat metric on $T^n$, and let $R$ be also Killing, and $\partial_{\varphi_i}$ be generators of the isometry group. So I asked: does $\mathcal{O}$ commutes with the isometry group?