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 $x(t)\in (\mathbb{Z}[t])^n$.
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$
If this equation has a polynomial solution in $(\mathbb{Z}/p\mathbb{Z}[t])^n$ 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.

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

Since there may be counterexample, you can add freely aditional condtions.

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