Find and determine the stationary points for z=x^3 -xy^2 + y^3

Hello, I am stuck trying to solve this problem at the point where you're supposed to solve the simultaneous equations of dz/dx = 0 and dz/dy = 0

I have got 3x^2 - y^2 and 2y - 2xy to solve (if these are correct) but i am struggling. Any help would be great :)

∇ f = {3x^2 - y^2, y (3y - 2x)}
∇ f = 0 ... has only one solution at (0,0)
... plug y = 2x/3 into 3x^2 - y^2 = 0 and x = 0 is the only solution

Answer: (0,0) is the only critical point