This paper deals with the Liapunov stability of the origin for the system x + xf(x) = 0, ÿ + yw(x) = 0, x, y ∈ R, f(0) > 0. (*) If there exists s(x, x) such that ys- ys 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
... [Show full abstract] exists at most one function w such that the system (*) admits ys - ys 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 (*) has the first integral ẏs- ys (an example is s(x, 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 (*) with w defined by s and f. We call such problems free coexistence-like. Some previous papers solved all the free coexistence-like problems except the one generated by s(x, x) =xx which is solved in this paper.