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.