Solution
= Solution
The equations are
$$
\ddot x+\alpha x^3+\beta xy^2=0,
\qquad
\ddot y+\alpha y^3+\beta x^2y=0.
$$
Hence
$$
\dot J=(\beta-\alpha)xy(x^2-y^2),
$$
which is not generally zero. Conservation requires \b[$\beta/\alpha=1$]. Then the potential is $\alpha(x^2+y^2)^2/4$, so the action has <rotational symmetry>. By <Noether's theorem>, $J$ is the conserved angular momentum, up to the sign convention used in its definition.