Inspired by this question on homothety vector field we ask the following question

Let $M$ be a manifold equiped by a volum form $\Omega$. With some abuse of terminologies we say: A strong constant divergence vector field is a vector field $X$ with the property that $\forall\; Y\in C(X)$ we have $Div_{\Omega}(Y)=0$. Here $C(X)$ is the centralizer of $X$.

An example of this strong situation is a Kronecker vector field on torus with the standard structure of torus: The vector field on torus tangent to $\partial_x+\theta \partial_y$ where $\theta $ is an irrational number.

Question: What is an example of a constant divergence vector field $X$ on a manifold $(M, \Omega)$ which is strongly constant divergence with respect to NO volum form on $M$?

Inspired by this question on homothety vector field we ask the following question

Let $M$ be a manifold equiped by a volum form $\Omega$. A strongly constant divergence vector field is a vector field $X$ with the property that $\forall\; Y\in C(X)$ we have $Div_{\Omega}(Y)=0$. Here $C(X)$ is the centralizer of $X$.

An example of this strong situation is a Kronecker vector field on torus with the standard structure of torus: The vector field on torus tangent to $\partial_x+\theta \partial_y$ where $\theta $ is an irrational number.

Question: What is an example of a constant divergence vector field $X$ on a manifold $(M, \Omega)$ which is strongly constant divergence with respect to NO volum form on $M$?