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 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)

A curvature description for center condition for quadratic vector field

An explicit formula for a flat metric compatible to certain polynomial vector field with center