Inspired by [this question on  homothety vector field](https://mathoverflow.net/questions/153581/does-for-every-vector-field-there-always-exist-a-volume-form-for-which-the-vecto) 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 $M$?