A functional equation involving comparable weighted quasi-arithmetic means

We discuss the functional equation $$f(M(x,y))=f(N(x,y)) \qquad (x,y\in I) $$ where I is a nonvoid open subinterval of the set of real numbers ?, f:?I??? is an unknown function, and M, N are weighted quasi-arithmetic means on?I. Our typical assumption will be that M and N are comparable. Complete solution will be presented under the additional supposition that f is the product of the generating functions of?M and?N, respectively.