Global Phase Portraits of Uniform Isochronous Centers System of Degree Six with Polynomial Commutator

This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form \(\dot x = - y + xf(x,\,y),\,\,\dot y = x + yf(x,\,y)\) $$\dot{x}=-y+xf(x,y),\dot{y}=x+yf(x,y)$$ , where f(x, y) = a₁x + a₂xy + a₃xy² + a₄xy³ + a₅xy⁴ = xσ(y), and any zero of 1 + a₁y + a₂y² + a₃y³ + a₄y⁴ + a₅y⁵, \(y = \bar y\) $$y=\overline{y}$$ is an invariant straight line. At last, all global phase portraits are drawn on the Poincaré disk.