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 $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\right)V\;\;\;\;\;(V)'$$  has  an isochronous center  at $(0,0)$.  

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

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


The  motivations are  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