Existence of integral of motion

Question

Consider the following dynamical system $$\begin{cases}\dot x = y-\varepsilon (x^2+y^2)x \\ \dot y=-x-\varepsilon(x^2+y^2)y\end{cases} $$ There exists some integral of motion (or constant of motion, that is a function which is constant on the solutions of the dynamical system) depending on $\varepsilon$?

Answer 1

Hint.

$$\begin{cases}x \dot x = x y-\varepsilon (x^2+y^2)x^2 \\ y\dot y=-xy-\varepsilon(x^2+y^2)y^2\end{cases} $$

and after addition

$$ \frac 12(r^2)' = -\varepsilon r^4 $$