The last free coexistence-like problem

This paper deals with the Liapunov stability of the origin for the system $$\ddot x+xf(x)=0,\quad \ddot y+yw(x)=0,\quad x,y\in\Rfi,\quad f(0)>0. \eqno(\star)$$ If there exists $s(x,\dot x)$ such that $\dot y s-y\dot s$ is a first integral, and some smoothness and nondegeneracy conditions hold, then the stability is equivalent to ``coexistence'' of periodic solutions of every Hill's equation in a certain family. Given the functions $s$ and $f$, there exists at most one function $w$ such that the system $(\star)$ admits $ \dot y s-y \dot s $ as first integral, but generally no such $w$ exists. Certain special functions $s$ have the property that $w$ can be found in connection with each $f$ so that $(\star)$ has the first integral $ \dot y s-y \dot s $ (an example is $s(x,\dot x)=x$ where we can choose $w=f$). Each of these special functions $s$ generates the following problem: determine all the functions $f$ such that the origin is a stable equilibrium for $(\star)$ with $w$ defined by $s$ and $f$. We call such problems \it free coexistence-like.\rm Some previous papers solved all the free coexistence-like problems except the one generated by $s(x,\dot x)=x\dot x$ which is solved in this paper.