Some central forces-stability (Q996106)

An ordinary differential equation (ODE) of the form \[ \ddot{x}=-xf(x,y),\quad \ddot{y}=-yf(x,y) \] with analytical \(f:\mathbb{R}^2\to \mathbb{R}\) is considered and the stability of the origin is studied. Under the assumption that this ODE admits a certain first integral, the ODE is reduced to \(\ddot{x}=-x g(xy)\), \(\ddot{y}=-y g(xy)\) where \(g:\mathbb{R}\to\mathbb{R}\) is some analytical function. It is shown that the origin of this ODE is stable if and only if \(g\) is constant.