A Pexider equation containing the aritmetic mean

In this paper we determine the solutions $(\varphi,f_1,f_2)$ of the Pexider functional equation $$\varphi\Big(\frac{x+y}2\Big)\big(f_1(x)-f_2(y)\big)=0,\qquad (x,y)\in I_1\times I_2,$$ where $I_1$ and $I_2$ are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of $\varphi$ is closed. We also discuss particular solutions where this condition is not met.