 Cubic swap-map two-cycle factorization

For $F(x,y)=(y,a(x-x^3/3)-y)$, a genuine two-cycle obeys $F(x,y)=(y,x)$ with $x\ne y$. Subtracting its two component equations gives $x^2+xy+y^2=3$, then substitution gives the displayed factorization. Cancelling $x+y$ loses the antipodal cycle $(\sqrt{3},-\sqrt{3})\leftrightarrow (-\sqrt{3},\sqrt{3})$, present for every $a>0$. The nonzero-sum family instead has $xy=3/a$ and $x+y=\pm \sqrt{3+3/a}$, requiring $a>3$. Retaining all vanishing factors is essential in polynomial cycle classification.