A non-geodesible foliation of $S^3$ or $S^2\times S^1$

Question

Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?

If the answer is affirmative, can one produce an example of a non-geodesible foliation of $S^3$ or $S^2\times S^1$ whose leaves are mapped to the singular foliation of $S^2$ associated to the Poincaré compactification of a planar quadratic vector field? (By "mapped" we mean the Hopf map $S^3\to S^2$ or the projection $S^2\times S^1\to S^2$.)

For detailed motivation please see the following post.

Answer 1

If a 1-dimensional foliation of a 3-manifold contains a Reeb component, then it is not geodesible. This follows from a theorem of Sullivan, see Section 2.5, Obstruction 1. A Reeb component is a saturated annulus embedded in the manifold with induced foliation looking like:

One can find a 1-dimensional foliation of any 3-manifold containing such a Reeb component. Foliate $T^2\times I = S^1 \times (S^1\times I)$ by foliating $S^1\times I$ by the Reeb foliation, and extending by the product with $S^1$ to $T^2 \times I$. Now, embed $T^2\times I$ in a manifold $M$ with this foliation, and extend it to a 1-dimensional foliation on the rest of $M$.