Assume that $X$ is a non vanishing vector field on torus $\mathbb{T}^2$. Is there a vector field $Y$ such that $[X,Y]$ does not vanish on torus?
What about if we replace torus by an arbitrary compact manifold of zero Euler characteristic?
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If I may assume $X$ to be expressible in a global coordinate system $\{x^1,x^2\}$ in which $X=\partial_1$ , (this is always possible locally as $X$ is non-vanishing) the answer to the first question is yes . For $Y = Y^1\partial_1+Y^2\partial_2$ to be a required field, one only needs to ensure that not both $\partial_1(Y^i)$ vanish simultaneously. For instance, let $Y=cos(x^1)\partial_1+sin(x^1)\partial_2$. As for the second question, the unit circle $S^1$ looks like a counter-example to me.
