Consider a differential equation of the form $$f^{(n)}(z)=P(z,f(z),f'(z),...,f^{(n1)}(z))$$ where $P$ is a polynomial in $n+1$ variables, with the initial condition (for definitness) $$f^{(k)}(0)=0,\qquad 0\leq k\leq n1.$$ By the ODE existence theorem, there exists a unique solution $f(z)$ which is holomorphic near the origin of the complex plane. My question is whether it is possible to decide whether $f(z)$ can be analytically continued to the whole complex plane. For example, suppose that the coefficients of P are rational numbers. Is there a finite decision procedure that will take P and determine whether $f(z)$ is entire?

One way to see that the local holomorphic function $f$ is entire is to see if the ODE defined by the polynomial $P$ is integrable. If its not, then you can assure that the solutions are not entire due to the fact that a nonintegrable ODE has solutions with singularities. In fact, this is the way to show nonintegrability for ODE, to find the singularities of some certain solutions. For an explicit example, you can search on the vast literature about the integrability question of the HenonHeiles system. Hope this is what you were searching. 

