Find the fixed points of the system

$$\begin{aligned} &\dot{x}=x(x+2 y-3) \\ &\dot{y}=y(3-2 x-y) \end{aligned}$$

Local linearization shows that all the fixed points with $x y=0$ are saddle points. Why can you be certain that this remains true when nonlinear terms are taken into account? Classify the fixed point with $x y \neq 0$ by its local linearization. Show that the equation has Hamiltonian form, and thus that your classification is correct even when the nonlinear effects are included.

Sketch the phase plane.