In the course of my research I came across the very simple-looking equation
$$ x = \text{sgn}(M x) \,, $$
where $x$ is a real $N$-dimensional vector and $M$ is a real symmetric $N \times N$ matrix. Because of the sgn, the solution set is contained within $\{-1, 0, 1\}^N$.
As a special case, if $M$ is the identity, then $\{-1, 0, 1\}^N$ is clearly the solution set, since the vector equation decomposes into $N$ independent scalar equations,
$$ x_i = \text{sgn}(x_i) \,, \qquad i = 1, ..., N, $$
each with solution $x_i \in \{-1, 0, 1\}$.
I am wondering if anything can be said about the solution set for general $M$, for example the size of the set. This is such a simple equation I'm sure it's been studied in many contexts before, but I don't know where to look or what name to search.