  Let $I\subseteq\mathbb{R}$ be a nonempty open subinterval. We say that a
two-variable mean $M:I\times I\to\mathbb{R}$ enjoys the \emph{balancing
property} if, for all $x,y\in I$, the equality \begin{equation}\tag{1}
M\big(M(x,M(x,y)),M(M(x,y),y)\big)=M(x,y) \end{equation} holds.
  The above equation has been investigated by several authors. The first
remarkable step was made by Georg Aumann in 1935. Assuming, among other things,
that $M$ is \emph{analytic}, he solved (1) and obtained quasi-arithmetic means
as solutions. Then, two years later, he proved that (1) characterizes
\emph{regular} quasi-arithmetic means among Cauchy means, where, the
differentiability assumption appears naturally. In 2015, Lucio R. Berrone,
investigating a more general equation, having symmetry and strict monotonicity,
proved that the general solutions are quasi-arithmetic means, provided that the
means in question are \emph{continuously differentiable}.
  The aim of this paper is to solve (1), without differentiability assumptions
in a class of two-variable means, which contains the class of \emph{Matkowski
means}.
