What is a classification of all quadratic vector fields

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

Extension of a vector field to an orthonormal frame for a flat metric

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