Consider a trivial fiber bundle $\mathbb{R} \times M \rightarrow M$ with a fundamental vector field $\xi$ running up the fibers. I would like to study the pair of differential equations
$$\mathcal{L}_{\xi} f_1 = F(x,f_1,f_2) $$
$$\mathcal{L}_{\xi} f_2 = G(x,f_1,f_2) $$
Here, $x$ are coordinates on the total space $\mathbb{R} \times M$, and assume that everything here is real. Note that $F,G$ are not in general homogeneous functions of $f_1$ or $f_2$. Under what conditions do global solutions $f_1,f_2$ exist?
My first instinct was Picard's theorem, but as far as I know, that doesn't necessarily apply unless I can decouple $f_1$ and $f_2$. After some googling I came upon the Cauchy–Kowalevski theorem, however that does not seem to imply global existence, which is necessary for my application. Is this problem too general to say anything about, or are there known results for such differential equations?
