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 $x(t)\in (\mathbb{Z}[t])^n$.  
$$
\begin{cases}
\dfrac{dx(t)}{dt}=V(x(t)),\\
x(0)=\mathbf 0\in \mathbb{R}^n
\end{cases}
$$  
**Quesstion**. If this equation has a polynomial solution in $(\mathbb{Z}/p\mathbb{Z}[t])^n$ for every sufficiently large prime $p$, then does the original equation have polynomial solution in $(\mathbb{Z}[t])^n$?

Here, $F\in (\mathbb{Z}/p\mathbb{Z}[t])^n$ is a solution of this equation, if 
$F$ is a formal solution of a differential equation with $\tilde{V}: (\mathbb{Z}/p\mathbb{Z})^n\rightarrow  (\mathbb{Z}/p\mathbb{Z})^n$ which is a reduction of $V$  mod $p$

Moreover what if $V$ is $p$-closed for each sufficiently large $p$ i.e. $V^p=0$ as derivation on $\mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$.  
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.

Full reference:<br>
Jean-Benoît Bost, "[Algebraic leaves of algebraic foliations over number fields](http://www.numdam.org/item/PMIHES_2001__93__161_0/)", Publications Mathématiques de l'IHÉS, Tome 93 (2001), pp. 161-221. [MR1863738](https://mathscinet.ams.org/mathscinet-getitem?mr=1863738), [Zbl 1034.14010](https://zbmath.org/?q=an:1034.14010).

Since there may be counterexamples, you can freely add additional conditions.  

I think this is a kind of non linear version of Grothendieck-Katz conjecture.