What is the number of connected components of solutions to $y^3 +3xy^2 - x^3 = 1$ 
What is the number of connected components of solutions to $y^3 +3xy^2 - x^3 = 1$

Attempt
$x^3 - 3y^2x + 1 - y^3 = 0$
$p := -3y^2$,   $q := 1 - y^3$
The discriminant is 
$Q = (p/3)^3 + (q/2)^2 = (-y^2)^3 + (1 - y^3)^2 / 8 = -y^6 + 1/8 -  2y^3 / 8 + y^6 / 8 = (-7y^6 - 2y^3 + 1/8) / 8$
Hence there exist $y$ such that $Q < 0$. They yield 3 real solutions and therefore 3 connected components.
But we need to prove the components do not intersect.

There are three disjoint components to the solution curves.
Consider the disjoint regions formed by the following half lines:
$1)$                      $$ \{ (x,0): x\ge 0\}$$
$2)$                      $$ \{(x,-x): x\le 0\}$$
$3)$                           $$ \{(0,y): y\le 0 \}$$ 
Each component is contained in one and only one of the  rgions.
Thus the three branches of the solution curve are disjoint.    
