A vector field with specified curl

Question

I need a vector field $\vec{F}:\mathbb{R}^3\to\mathbb{R}^3$ such that $$\mathrm{curl}\ \vec{F}(x,y,z) \cdot \left(\frac{-x}{\sqrt{x^2+1}},\ 0,\ \frac{1}{\sqrt{x^2+1}}\right) = 1.$$ (This equality, of course, holds if $\mathrm{curl}\ F(x,y,z) = \left(\frac{-x}{\sqrt{x^2+1}},\ 0,\ \frac{1}{\sqrt{x^2+1}}\right)$. But this is not necessarily the case.) Can somebody help me?