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](https://www.sciencedirect.com/science/article/pii/0022039686901130), 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:

https://mathoverflow.net/questions/271543/does-this-function-belong-to-l2-mathbbd


https://mathoverflow.net/questions/164059/codimension-of-the-range-of-certain-linear-operators