CZ-SK-MO 2008 Problem 1

Find all pairs of real numbers $(x,y)$ satisfying: $x+y^2=y^3,$ $y+x^2=x^3.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.