Let $M $ be a manifold, and $\sf X, \sf Y$ vecor fields with the same set of zeros: i.e., for every $x\in M$, $\mathsf{X}(x)=0$ iff $\mathsf{Y}(x)=0$. Denote $V=\{ x\in M \mid \mathsf{X}(x)=0 \}$.



> 
> Is there a diffeomorphism $\varphi:M\to M$ such that $\varphi\_{\*}\sf X = \sf Y$? 
>  What if $\varphi$ is required to fix $V$ pointwise?
> 
> 
> 



Is the action of diffeomorphisms transitive on the set of vector fields with prescribed zero locus?
Let me summarize the conclusions in the comments. This is rarely true:


There are local obstructions around fixed points: For example the order of vanishing (Andreas Cap), or the Hopf index of an isolated equilibrium (more generally the Conley index) must be preserved by the diffeomorphism. 


There are also global obstructions. A simple observation is the following: The diffeomorphism must map periodic orbits to periodic orbits. Hence the cardinality of the set of periodic orbits must be the same. The irrational flow on the two torus cannot be mapped to the rational flow (where all orbits are periodic). 


Locally away from the equilibria the statement is true. There is a standard form around such points: This is called the flow box theorem. See for example Theorem 2 [here](http://www.math.usm.edu/lambers/mat605/lecture20.pdf).


