Solution
= Solution
Along a trajectory, the <chain rule> gives
$$
\frac{dH}{dt}=2x\dot x+(2y-4y^3)\dot y
=2x(y-2y^3)-(2y-4y^3)x=0.
$$
Thus $\boxed{H=x^2+y^2-y^4}$ is a <first integral>, and every trajectory lies on one of its <level sets>.
= Solution
Along a trajectory, the <chain rule> gives
$$
\frac{dH}{dt}=2x\dot x+(2y-4y^3)\dot y
=2x(y-2y^3)-(2y-4y^3)x=0.
$$
Thus $\boxed{H=x^2+y^2-y^4}$ is a <first integral>, and every trajectory lies on one of its <level sets>.