Extension of a generalized Pexider equation

The equations $k(s+t)=\ell(s)+n(t)$ and $k(s+t)=m(s)n(t)$, called Pexider equations, have been completely solved on $\mathbb{R} ^2.$ If they are assumed to hold only on an open region, they can be extended to $\mathbb{R} ^2$ (the second when $k$is nowhere 0) and solved that way. In this paper their common generalization $k(s+t)=\ell(s)+m(s)n(t)$ is extended from an open region to $\mathbb{R} ^2$ and then completely solved if $k$ is not constant on any proper interval. This equation has further interesting particular cases, such as $k(s+t)=\ell(s)+m(s)k(t)$ and $k(s+t)=k(s)+m(s)n(t),$ that arose in characterization of geometric and power means and in a problem of equivalence of certain utility representations, respectively, where the equations may hold only on an open region in $\mathbb{R} ^2.$ Thus these problems are solved too.