This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).
Linear vector field are always complete vector field.
But for higher order polynomial vector field, I guess that the solutions which are not a complete orbits, are not in $\ell^2$. My motivation is that according to a Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator. But your question is very interesting for me since it implicitelty suggest to consider some different function space for consideration of $D_f$ the derivational operator associated to the vector field.