IR-TST 2012 Problem 15

Find all integer numbers $x$ and $y$ such that: $(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).$

Proposed by Mahyar Sefidgaran

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.