BG-MO 1966 Problem 1

Prove that the equation $3x(x-3y)=y^2+z^2$ doesn't have any integer solutions except $x=0,y=0,z=0$ .

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.