Find the most general cubic form

$$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$$

which satisfies Laplace's equation, where $a, b, c$ and $d$ are all real. Hence find an analytic function $f(z)=f(x+i y)$ which has such a $u$ as its real part.