Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For concreteness in the image below imagine a continuous vector field throughout space that is tangent to the tangled curves.

To be specific suppose there are a finite number of $S^1$ curves embedded in the torus $T^3$. Under what circumstances can I choose a continuous vector field in $T^3$ that is everywhere non-zero and tangent to the embedded curves. Equivalently when can I extend the curves to a complete foliation of $T^3$ by one-dimensional curves?

If it is not possible to extend to a non-zero vector field, is there anything I can say about the topology of the submanifolds where the vector field is zero?

EDIT: I think this is related to the question of whether the three-dimensional submanifold $S\subset T^3$ which is the complement of the embedded curves has a trivial tangent bundle. However I don't know if this is a sufficient condition, and I don't know if three dimensional manifolds with trivial tangent bundles have been classified in some way.