AT-OEMO-FED1 2005 Problem 2

For how many integers $a$ with $|a| \leq 2005$ , does the system $x^2=y+a$ $y^2=x+a$ have integer solutions?

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.