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 this old paper 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**

-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 $|\partial h/\partial\theta|^2+|\partial h/\partial\zeta|^2$ is less than $a^2+b^2$, then you can use Moser's trick (nice discussion of it here http://concretenonsense.wordpress.com/2009/09/03/symplectic-geometry-ii/) to prove the answer is yes. In particular, you can show that $dh+ad\theta+bd\zeta$ is strongly isotopic to $a d\theta+bd\zeta$.