Non-zero, divergence-free vector fields on 2-torus Suppose $X$ is a nowhere vanishing vector field on the 2-torus that preserves the standard area element $\mu=d\theta\wedge d\zeta$. By area preservation, $$ i_X\mu=dh+ad\theta+bd\zeta, $$ for some smooth function $h$ and constants $a,b\in\mathbb{R}$. Is there a diffeomorphism $\phi$ of the 2-torus such that $\phi^*X$ is a (re-scaling of) the constant vector field $Y=b\frac{\partial}{\partial\theta}-a\frac{\partial}{\partial\zeta}$?
Based on the discussion in the old paper On the Measure-Preserving Flow on the Torus by T. Saito
it seems like the answer definitely could be yes, but I'm having a hard time proving it myself or finding a reference that addresses the question.
Progress so far

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*When one of the constants, say $b$, is zero (note that both cannot be zero) the answer is yes. In this case one such $\phi$ is $\phi^{-1}(\theta,\zeta)=(\theta+\frac{1}{a}h(\theta,\zeta),\zeta)$.


*When the maximum value of $\lvert\partial h/\partial\theta\rvert^2+\lvert\partial h/\partial\zeta\rvert^2$ is less than $a^2+b^2$, then you can use Moser's trick (nice discussion of it here in lewallen's post "Symplectic nonsense II" on Concrete Nonsense) to prove the answer is yes. In particular, you can show that $dh+ad\theta+bd\zeta$ is strongly isotopic to $a d\theta+bd\zeta$.
 A: The orbits of the flow by the vector field $X$ forms a foliation $\mathcal{F}_X$ of $T^2$. There is a transverse measure to the foliation: for a curve $\sigma$ transverse to $\mathcal{F}_X$, define the measure of $\sigma $ to be $\int_\sigma i_X\mu$. Since the vector field $X$ and 2-form $\mu$ are preserved by the flow by $X$, this measure is invariant under the flow by $X$, and in fact by transverse isotopy to $\mathcal{F}_X$ rel endpoints. 
I think it's well-known that a measured foliation $\mathcal{F}_X$ is homeomorphic to a foliation by lines of a fixed slope (in your case, it would be slope $a/b$). However, I don't know what regularity one can choose for this homeomorphism, in particular, is it a diffeomorphism?  
Addendum: For one description of measure foliations, you can have a look at A primer on mapping class groups. However, I think the point I'm making is fairly simple. Consider a simple closed curve $\sigma$ transverse to $\mathcal{F}_X$. The curve $\sigma$ has a measure (absolutely continuous with respect to Lebesgue measure) coming from the transverse measure to $\mathcal{F}_X$. Every leaf of $\mathcal{F}_X$ must meet $\sigma$, and cutting $T^2$ along $\sigma$ gives an annulus, with a foliation consisting of intervals connecting both sides. This must be a product foliation, and the identification of opposite sides rotates $\sigma$ by some fraction $\alpha$ (since the flow from one side to the other preserves transverse measure). So take a Euclidean annulus $A$ with the same area at $T^2$, and with geodesic boundary components of the same length as the measure of $\sigma$. There is a canonical way to connect opposite sides by orthogonal lines, so glue opposite sides by an $\alpha$-fraction rotation of the circle. Then I think there is homeomorphism sending this torus to the original $T^2$, and sending the foliation by lines orthogonal to $\partial A$ to $\mathcal{F}_X$. Being a bit more careful, I think one modify this map to be an area-preserving map, but I haven't thought this through carefully. 
