RO-NMO 2021 Problem 104

Determine all nonzero integers $a$ for which there exists two functions $f,g:\mathbb Q\to\mathbb Q$ such that $$f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.$$ Also, determine all pairs of functions with this property.

Vasile Pop