closed integral curves and conservative fields If a vector field $\vec{E}(\vec{x})$ in $\mathbb{R}^3$ has no closed integral curves, does this imply that the field is conservative, i.e. there is some scalar function $V(\vec{x})$ such that $\vec{E}(\vec{x}) = -\nabla V(\vec{x})$?

No, consider the vector field $\mathbf{E}(x,y,z)=\left< 0, 1 + x^2, 0 \right>$. You can check that this has non-zero curl and cannot be a gradient, and has no closed integral curves.
