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 $\varepsilon$? If not why?
I tried building (with "hands") an Hamiltonian in a right way (that is a function which derivatives with respect to $x$ and $y$ are the r.h.s. of my system...), without success. What I expected was to find none integral of motion. The original problem was a counter example of the theorem of Poincare about the continuation of the periodic orbits: if a was able to show an integral of motion (depending on $ε$), I could state the existence of a periodic orbit for the perturbed system, "near" the periodic orbit of unperturbed system: $x(t)=\cos t$ and $y(t)=\sin t$. But using the Lyapunov function $V=x^2+y^2$, one can see that there is no periodic orbit: the origin is an asymptotically stable equilibrium point.