Dynamics of the Painlevé-Ince equation (Q2095509)

The Painlevé-Ince differential equation can be rewritten as the system \[ \frac{{dx}}{{dt}} = -3xy-y^3, \quad \frac{{dy}}{{dt}} = x \tag{1} \] having the first integral \[ H(x,y) = \frac{(x^2+y^2)^2}{2x+y^2}. \] The authors determine the phase portrait of (1) in the Poincaré disk. It has a continuum of homoclinic orbits to the equilibrium at the origin and a continuum of heteroclinic orbits connecting the origin with an equilibrium point at infinity.