In this paper we deal with attractive central forces, more precisely with the system \[\ddot x = -xf(x,y),\ \ \ddot y = -yf(x,y),\ \ f(0,0) > 0\ \ f\in\mathcal{C}^\omega.\]
We characterize the stability of the origin whenever the system
admits a first integral of the following kind
\[V(x,y, \dot x, \dot y) = u(\dot x, \dot y) + \Pi (x,y),\]
with $u(\dot x, \dot y) = a\dot x^2 + b\dot x\dot y + c\dot y^2$
undefinite and make repairs to some optimisms we have committed in
\cite{3}.