I posted this topic on StackExchange, but it may suit this forum better.
Consider a bundle $(E,\pi, M)$ and let $k\in \mathbb N$. I am going to adopt the notations and conventions by Saunders.
Preliminaries
A first-order jet field on $\pi$ is a section of the bundle $(J^1\pi,\pi_0^1, E)$, $\pi_0^1$ being the target projection, i.e. $\pi_0^1(j^1_m\phi)=\phi(m)$. More generally, a $k$-th order jet field on $\pi$ is a section of the bundle $(J^k\pi, \pi_{k-1}^k, J^{k-1}\pi)$, $\pi_{k-1}^k$ being the projection $j^k_m\phi\mapsto j^{k-1}_m\phi$ (Actually, Saunders gives only the definitions for $k=1,2$).
A first-order jet field $X$ on $\pi$ is termed integrable if there is a local section $\psi$ of $\pi$ such that $X\circ \psi=j^1\psi$. Such a section is called an integral section of $X$.
There is a canonical embedding: $$ \iota_{k}\colon j^{k}_m\phi\in J^{k}\pi\to j^{1}_m(j^{k-1} \phi)\in J^1(\pi_{k-1})\,, $$ where $\pi_{k-1}\colon J^{k-1} \pi\to M$ is the $(k-1)$-source projection on $\pi$, i.e. $\pi_{k-1}(j^{k-1}_m\phi)=m$.
Question Let $X$ be a $k$-th order jet field on $\pi$. Then $X_k=\iota_k\circ X$ is a first-order jet field on $\pi_{k-1}$. If $X_k$ is integrable and $\psi$ is an integral section of $X_k$, is there a local section $\phi$ of $\pi$ such that $\psi=j^{k-1}\phi$? (Saunders proves it only for $k=2$ on page 181).
Also, I would appreciate your references to higher-order jet fields and integral sections.
Many thanks for considering my requests.
