Phase portrait of x dot equals two x times y minus a ($\dot{x}=2x(y-a),\dot{y}=1-x^2-y^2$)

This reflection-symmetric planar system has equilibria $(0,\pm 1)$ and, for $|a|\le 1$, $(\pm \sqrt{1-a^2},a)$. Its divergence is the constant $-2a$, excluding periodic orbits when $a\ne 0$ by the Bendixson-Dulac criterion. At $a=0$ it is Hamiltonian withand the nonzero equilibria are nonlinear centers. At $a=\pm 1$ the symmetric equilibria undergo pitchfork bifurcations.