I am having trouble "visualizing" the behavior of the two objects mentioned in title. And the question I'm raising might be vague in some sense (or too specific). Also I am hope for some reference related and accessible for a physics student. Another question that I asked a few days ago related is HERE

I see that it is rather easy for a manifold to have an almost contact structure (ACS), obstruction being 3rd integral S-W class. However, it is not always possible to find one for any given nowhere-vanishing vector fields (or a 1-form), trivial example being $S^1 \times S^4$, with vector field (or 1-form) pointing along $S^1$.

So, ACS only select some of 1-forms.

**(1) So I am wondering if there is some characteristic property of such 1-forms?**

To me, the 1-form along $S^1$ of $S^1 \times S^1$ and $S^1 \times \mathbb{CP}^2$ doesn't have much difference (they are both very nice), but they have different fate. This is one reason that puzzles me.

Also, **(2)can some seemingly "problematic" vector fields, say, the ones generating irrational flows, correspond to some ACS?**

Finally, on some recent papers on 5d gauge theory and Killing spinors <Twisted supersymmetric 5D Yang-Mills theory and contact geometry>, <Supersymmetric Gauge Theories on the Five-Sphere>, among others, a concept "**contact instanton**" arises: a (anti-) contact instanton is defined through equation

\begin{equation} *(\kappa\wedge F) = \pm F \end{equation} where $F$ is field strength of some gauge field. $\kappa$ is contact 1-form of certain K-contact structure.

It seems that the equation only uses a unit-length-nowhere-vanishing 1-form $\kappa$, without using further information from contact structure. **(3)So I am wondering if such concept makes sense (or interesting) for any given nowhere-vanishing $\kappa$? (worrying about those "problematic" 1-forms) Is this kind of object studied in math community?**

Thank you!