Let M be a Riemannian Manifold, $X$ is a smooth vector field on M with isolated zeros.
Is there a one-form $\omega$ with isolated zeros such that $\omega(X)$ has nontrivial zeros? (nontivial zero means that the piont is neither in $X$'s zeros nor in $\omega$'s zeros.)
If this $\omega$ exist, how to construct it? 



A question about a one-form on Riemannian manifold
Assuming the dimension of $M$ is at least 2 (otherwise it's false), you can do the following. Let $p\_1,p\_2,\dots$ be isolated points where $X$ does not vanish but where you want $\omega$ to vanish. In a neighborhood $U\_i$ of each $p\_i$, there are coordinates $(x^1,\dots,x^n)$ centered at $p\_i$ on which $X$ has the coordinate representation $X = \partial/\partial x^1$. In each $U\_i$, let $\omega\_i = dx^2 + |x|^2 dx^1$. Then let $U\_0$ be the complement of {$p\_1,p\_2,\dots$}, and let $\omega\_0=X^\flat$ (the 1-form dual to $X$ via the metric). Let {$\phi\_0,\phi\_i$} be a partition of unity subordinate to the cover {$U\_0,U\_i$}, and let $\omega = \sum\_{i\ge 0}\phi\_i\omega\_i$. The fact that $\omega\_i(X)>0$ at points other than $p\_i$ and zeros of $X$ ensures that $\omega(X)$ vanishes only at such points.


