<s>[INST] 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.
 [/INST] 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.    
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