The problem of the stability of the origin for the sytem (∗) x¨+xf(x)=0, y¨+yw(x)=0, f(0)>0, f∈C1, w∈C0, is said to be related to coexistence if (∗) has a first integral of the form y˙s(x,x˙)−ys˙(x,x˙). In this case the author says that (f,w,s) is coexistence-like. In this paper the author assumes that s is given by s(x,x˙)=x˙(1+αx), α∈R, and determines and constructs all the maps f such that the origin is a stable equilibrium for the system in (∗) with w(x)=(f(x)+xf′(x)+αx(4f(x)+xf′(x)))/(1+αx).
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