Are there any good reference to tackle the problem below?
Or, are there any know result?

Problem  
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector field on $\mathbb{R^n}$ defined by each $f_i$.

Let's consider a initial value problem for $x(t):\mathbb{R}\rightarrow \mathbb{R^n}$.  
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$  
If this equation has a polynomial solution in $\mathbb{Z}/p\mathbb{Z}[x]$ for every sufficiently large prime $p$,

Then does the original equation have polynomial solution in $\mathbb{Z}[x]$?

Moreover what if $V$ is $p$-closed for each sufficiently large $p$ i.e. $V^p=0$ as derivation on $\mathbb{Z}/p\mathbb{Z}$.  
Bost's paper about algebraicity of foliaiton seems related to this question but I don't know whether I can I apply techniques used there.

http://www.numdam.org/item/PMIHES_2001__93__161_0/

Since there may be counterexample, you can add freely  aditional condtions.  
Other keyword  
Grothendiek Katz conjecture