This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation).
Linear vector fields are always complete vector field so they does not satisfy your condition.
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 an interesting Paper of Chicone and Sotomayor, the solutions escape at infinity very fast(exponentially) since there is a hyperbolic singularity at equator.
On the other hand your question is very interesting for me since it implicitly suggests to consider some different function spaces to be acted by $D_f$, the derivational operator associated to the vector field $f$.
The motivations for study of this derivational operator is explained in the following posts: