The most horrible case- bash ever: Letting out the trivial case where $x = y =0$, we do this: $$3x^2y^2-y^4 = xy = x^4 - 3x^2y^2$$which gives: $$(x^2 -y^2 -2xy)(x^2 - y^2+ 2xy) = 0 \hspace{1cm} --------(*)$$Case 1: $$x^2 -y^2 -2xy = (x+y - \sqrt{2}y) (x+y +\sqrt{2}y) = 0$$This gives: $$x = (\sqrt{2}-1)y \hspace{0.15cm} \text{or} \hspace{0.15cm} x = -(\sqrt{2}+1)y$$Case 2: As $(*)$ was symmetric in $x$ and $y$, this case just swaps x and y from the above solution, so: $$y = (\sqrt{2}-1)x \hspace{0.15cm} \text{or} \hspace{0.15cm} y = -(\sqrt{2}+1)x$$Case 3: Both Case 1 and Case 2 are valid. This is easily dismissed. We plug in all the $x-y$ relations we found in the mother equation and get really ugly stuff. At the end of our misery, these are all 5 solutions we are getting: $$ (x,y) = (0,0), (\sqrt{\frac{2-\sqrt{2}}{4}},\sqrt{\frac{2+\sqrt{2}}{4}}), (-\sqrt{\frac{2-\sqrt{2}}{4}},\sqrt{\frac{2+\sqrt{2}}{4}}), (\sqrt{\frac{2-\sqrt{2}}{4}},-\sqrt{\frac{2+\sqrt{2}}{4}}), (-\sqrt{\frac{2-\sqrt{2}}{4}},-\sqrt{\frac{2+\sqrt{2}}{4}})$$

MathBoy23 wrote: The most horrible case- bash ever: Letting out the trivial case where $x = y =0$, we do this: $$3x^2y^2-y^4 = xy = x^4 - 3x^2y^2$$which gives: $$(x^2 -y^2 -2xy)(x^2 - y^2+ 2xy) = 0 \hspace{1cm} --------(*)$$Case 1: $$x^2 -y^2 -2xy = (x+y - \sqrt{2}y) (x+y +\sqrt{2}y) = 0$$This gives: $$x = (\sqrt{2}-1)y \hspace{0.15cm} \text{or} \hspace{0.15cm} x = -(\sqrt{2}+1)y$$Case 2: As $(*)$ was symmetric in $x$ and $y$, this case just swaps x and y from the above solution, so: $$y = (\sqrt{2}-1)x \hspace{0.15cm} \text{or} \hspace{0.15cm} y = -(\sqrt{2}+1)x$$Case 3: Both Case 1 and Case 2 are valid. This is easily dismissed. We plug in all the $x-y$ relations we found in the mother equation and get really ugly stuff. At the end of our misery, these are all 5 solutions we are getting: $$ (x,y) = (0,0), (\sqrt{\frac{2-\sqrt{2}}{4}},\sqrt{\frac{2+\sqrt{2}}{4}}), (-\sqrt{\frac{2-\sqrt{2}}{4}},\sqrt{\frac{2+\sqrt{2}}{4}}), (\sqrt{\frac{2-\sqrt{2}}{4}},-\sqrt{\frac{2+\sqrt{2}}{4}}), (-\sqrt{\frac{2-\sqrt{2}}{4}},-\sqrt{\frac{2+\sqrt{2}}{4}})$$ Shouldn't the solutions be flipped?