Let $Q^n$ be a closed manifold, $M = TQ$ its tangent bundle, $\xi$ be a differential equation on $M$ that satisfies the "canonical flip on $TTQ$" (a "second-order differential equation on $Q$"), but suppose $\xi$ is the zero vector at some point(s) of $M$.
If we define $p \sim q$ if there is a flow line $\Phi_t(g)$ of $\xi$ with $\Phi_{t_1}(g) = p$ and $\Phi_{t_2}(g) = q$, do the equivalence classes form a 1-dimensional foliation $\mathscr{F}$ of $M$?
If the equivalence classes form a 1-dimensional foliation of $M$, when is there a codimension-1 foliation $\mathscr{G}$ transverse to $\mathscr{F}$?
If there is a codimension-1 foliation $\mathscr{G}$ of $M$ transverse to $\mathscr{F}$, if $L_g$ is any leaf of $\mathscr{G}$, is $\{K_g\ |\ g \in L_g\} = \mathscr{F}$, where each $K_g = \{p \in M\ |\ \exists\ t \in \mathbb{R} \text{ with } \Phi_t(g) = p\}$?
Thanks in advance.