How to solve this difficult system of equations? $$1+4\lambda x^{3}-4\lambda y = 0$$
$$4\lambda y^{3}-4\lambda x = 0$$
$$x^{4}+y^{4}-4xy = 0$$
I can't deal with it. How to solve this?

Hint: It is obvious (see I eq.) that $\lambda \ne 0$. Then from II eq. we have $x=y^3$. Now write this in III eq. and find possible values for $y$ and then $x$ and $\lambda$.
