What  is  a  classification of  all quadratic  vector  fields

 $$\begin{cases}
x'=P(x,y)\\
y'=Q(x,y)
\end{cases}\;\;\;\; (V)$$
with  a  center  at origin such that $(\frac{x^2+y^2}{yP(x,y)-xQ(x,y})V$  has  an isochronous center  at $(0,0)$.  

Here $P,Q$  are  degree  $2$ polynomials.

In the  linear center, the only possibility?

In  particular does $y\partial_x-(x+x^2)\partial_y$ satisfy the  above  property?


The  motivations  is  mentioned in the  following  very helpful comment   by  Prof. Goodwillie  and the next two  posts:( The  role of  isochronous center is very essential. We realize of this importance after  this very helpful comment)

https://mathoverflow.net/questions/278166/extension-of-a-vector-field-to-an-orthonormal-frame-for-a-flat-metric/278173#comment685996_278173

https://mathoverflow.net/questions/277481/a-curvature-description-for-center-condition-for-quadratic-vector-field?noredirect=1&lq=1



https://mathoverflow.net/questions/279625/an-explicit-formula-for-a-flat-metric-compatible-to-certain-polynomial-vector-fi