On the functional equation Gx,Gy,x=Gy,Gx,y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) $$\end{document} and means

We consider the functional equation Gx,Gy,x=Gy,Gx,y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) $$\end{document}, posed in Jarczyk and Jarczyk (Aequ Math 72:198–200, 2006). We show that every continuous and reducible solution generates a mean resembling the weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation. This fact particularly implies that the equation is not a direct consequence of the bisymmetry equation and the reflexivity condition. The closedness of the family of solutions with respect to conjugacy is noted. Finally, the translative solutions, homogeneous solutions, and suitable iterative composite functional equation for single variable functions are discussed.