Isochronization of quadratic vector fields with center

Question

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

Answer 1

The rescalling $(V')$ of $(V)$ as described in the question has always an isochronous center when $(V)$ is a quadratic system with center.

The reason is that $d\theta(V')=1$ where $d\theta=(\frac{1}{x^2+y^2})(ydx-xdy)$.

Remark: The same argument as above can be applied to show the following:

Let $V$ be an arbitrary quadratic vector field on the plane and $(V')$ be the corresponding rescalling as in the question. (We no longer assume that $V$ has necessarily a center). Then all closed orbits or limit cycles of $V'$ which surround the origin have the same length provided we choose a Riemannian metric whose frame is in the form $\{V', f(x\partial_x+y\partial_y)\}$ where $f$ is an arbitrary positive smooth function.