On solvable operators which imply the general solution of a polygonal functional equation on the real plane (Q1888662)

As a way to answering a hitherto unsolved problem of M. A. McKiernan (unpublished) concerning the functional equation \(f(x+t,y)+f(x-t,y)+f(x,y+t)+f(x,y-t)=f(x+t,y+t)+f(x-t,y+t)+f(x+t,y-t)+f(x-t, y-t)\;(x,y,t\in\mathbb{R})\) the authors offer the following. Let \(F\) be the set of all functions \(f: \mathbb{R}^2\to\mathbb{R}\) and \(A\) the algebra of linear operators on \(F,\;S\subset A\) the commutative subalgebra with unit, generated by finite linear combinations of shift operators \(X^t, Y^t\) on \(F\;((X^t f)(x,y)=f(x+t,y),\;(Y^t f)(x,y)=f(x,y+t)).\) Then the ideal generated by \(\{X^t+X^{-t}+Y^t+Y^{-t}-(X^t+X^{-t})(Y^t+Y^{-t}) \mid t\in\mathbb{R}\}\) contains the operators \(\{(X^t-1)^7\mid t\in\mathbb{R}\},\;\{(Y^t-1)^7\mid t\in\mathbb{R}\}.\)