  The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is defined in terms of a
combination of the following two operations: addition, $a \oplus b :=
\max(a,b)$, and multiplication, $a \otimes b := a + b$. In this study, we
propose a new method to characterize the set of all solutions of a max-plus
two-sided linear system $A \otimes x = B \otimes x$. We demonstrate that the
minimum ``min-plus'' linear subspace containing the ``max-plus'' solution space
can be computed by applying the alternating method algorithm, which is a
well-known method to compute single solutions of two-sided systems. Further, we
derive a sufficient condition for the ``min-plus'' and ``max-plus'' subspaces
to be identical. The computational complexity of the method presented in this
study is pseudo-polynomial.
