Invariant Translative Mappings and a Functional Equation

Let \({K,M,N : \mathbb{R}^{2} \rightarrow \mathbb{R}}\) be translative functions. Then K is invariant with respect to the mapping \({(M,N) : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}\) if and only if the functions \({h = K(\cdot, 0), f = M(\cdot, 0), g = N(\cdot, 0)}\) satisfy the functional equation $$h(x) = h(f(x) - g(x)) + g(x),\,\, x\in \mathbb{R}.$$ If K, M, N are means, then h(0) =  f(0) =  g(0) = 0. The formal power solutions and analytic solutions of this functional equation, satisfying these initial conditions, are considered.