ContourPlot finds three solutions

ContourPlot[y - y^3 + 3*y*x^2 == 0, {x, -10, 10}, {y, -10, 10},FrameLabel -> {x, y}]

one of them is the solution you are looking for!

select the "right" curve:

cond = Reduce[ y - y^3 + 3*y*x^2 == 0, y] /. Or -> List 
(*{y == 0, y == -Sqrt[1 + 3 x^2], y == Sqrt[1 + 3 x^2]} *)
cond /. {x -> 0, y -> 0}

ContourPlot finds three solutions

ContourPlot[y - y^3 + 3*y*x^2 == 0, {x, -10, 10}, {y, -10, 10},FrameLabel -> {x, y}]

one of them is the solution you are looking for!

select the "right" curve:

cond = Reduce[ y - y^3 + 3*y*x^2 == 0, y] /. Or -> List 
(*{y == 0, y == -Sqrt[1 + 3 x^2], y == Sqrt[1 + 3 x^2]} *)
cond /. {x -> 0, y -> 0}
(*{True, False, False}*)

y == 0 is the solution!