The Chisini mean revisited

We investigate the $n$-variable real functions $\G$ that are solutions of the functional equation $\F(\bfx)=\F(\G(\bfx),\ldots,\G(\bfx))$, where $\F$ is a given function of $n$ real variables. We provide necessary and sufficient conditions on $\F$ for the existence and uniqueness of solutions. When $\F$ is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. Such solutions, called Chisini means, are then thoroughly investigated