Enneper surface is not injective I'm having trouble proving the following statement:


  $x(u, v) = (u − u^ 3/ 3
 + uv^2 , v − v^ 3/ 3
 + u^ 2 v, u^2 − v^ 2 )$ is a minimal surface and x is not injective


Proving that $x(u,v)$, which is also known as the Enneper surface, is minimal is not a problem. However, I can't prove that $x$ is not injective.

Is there any smart way to do this rather than trying some couples $ (a,b)$ and (c,d) hoping that $x(a,b)$ will be equal to $x(c,d)$ with $(a,b)$ and $(c,d) $ different couples?
One way is to look for symmetries. The second and third components are even functions of $u$, while the first is an odd function of $x$. So, if $(u_0, v_0)$ is a point such that the first component of $x$ is zero, then $x(u_0,v_0)=x(-u_0,v_0)$.

It's not hard to find a solution of $u−u^3/3+uv^2=0$ with nonzero $u$: for example, choose $v=0$ and solve for $u$.
Just use the counter example...  i.e.
$$x( \sqrt3 , 0) = (0,0,3)= x(- \sqrt3 , 0)$$
With
$$( \sqrt3 , 0) \ne (- \sqrt3 , 0)$$