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

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


The  motivations  is  mentioned in the  following two  posts:

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