If I would ask for $\phi'(x) = f( \phi(x))$ and $\phi(0)=f(0)$, I would get that the inverse of $\phi$ is forced to be of the form:


$$\phi^{-1}(z) = \int\_{0}^z \frac{1}{f(x)} d x.$$


Now it is natural to ask whether there is something similiar for the problem $\phi''(x) =f( \phi(x),\phi'(x))$? 


Or a little bit mor restrictive:
Given a (homogenous) polynomial $f\_j(X\_0, \dots X\_n) \in \mathbb{C}[X\_0, \dots, X\_n]$, where $j=1, \dots, m$, such that the variety $V(f\_1, \dots, f\_m)$ is a non singular variety over $\mathbb{C}$.


Q1: Is it known in general, if there exists a function $\phi: \Omega \rightarrow \mathbb{C}$, such that 
$$f\_j( \phi(s), \phi'(s), \dots, \phi^{(n)}(s))=0$$
or perhaps the easier problem with isolated variables
$$\left( \phi^{(n+1)}(s) \right)^{\mathrm{deg} (f\_j)}= f\_j( \phi(s), \phi'(s), \dots, \phi^{(n)}(s)).$$
for all $s \in \Omega$? Here $\Omega$ is any open subset of $\mathbb{C}$. Are there global meromorphic solutions? What is the algebraic invariant, which counts the linear independent solutions here?


Q2: (see Denis Serre's answer): What is known in the case $m=1$? Or what if require all $f\_j$'s to have the same degree?


Q3: Is there a general theory/algorithm, how to compute the solutions in special cases?


Q4: Are there some easy examples, which give an intuition, what we can expect to be true and what not? 


I make this community wiki, since I do not know wheter my question is naive. I have little intuition about this stuff. Please comment if you do some changes. As a motivation, the Weierstrass $\wp $ function does the job for elliptic curves.



Solving nonlinear ODE's
The answer is **NO** in general because your differential system is overdetermined: it has several equations and only one unknown function. Take the polynomials $\omega^2 X\_0-X\_2$ and $X\_2^2-X\_1^2-X\_0^2$ with $\omega^4\ne1$. The solutions of the first equation are $ae^{\omega s}+be^{-\omega s}$, but none of them satisfy the second equation, but $\phi\equiv0$. 


A more interesting question should be which varieties $V$ yield non-trivial solutions. The answer must be of geometrical nature. Overdetermined linear systems have been studied for a long time (dating back to Goursat and E. Cartan), but the situation for nonlinear ones might be widely open.


