What is an example of a gradient vector field $X$ on a Riemannian manifold $(M,g)$ which cannot be converted to a divergence free vector field via the following processes:

First we remove the singularities $S$ from $M$ then we set $M:=M\setminus S$

We are allowed to reparameterize $X$ to $X:=fX$ for some positive function $f$

We are allowed to change the initial Riemannian metric $g$ to a new metric $g'$ for computation of divergence of $X$ with respect to this new $g'$ to obtain a vector field $X$ on $M$ with $\operatorname{div}_{g'} X=0$.

In other words, with some abuse of terminology, we ask: *"Is every gradient vector field a divergence free vector field?"*

An obvious example is $X=x\partial_x+y\partial_y$ is a divergence free vector field on the punctured plane after rescaling $X:=\frac{1}{x^2+y^2}X$

Is the answer yes, at least in low dimensions?

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