A question on $x \pmod 1 \in \Bbb R^n$?

A question on $x \pmod 1 \in \Bbb R^n$?

Let $x \pmod 1 \in \Bbb R^n$ be the fractional portion of $x \in \Bbb R^n$.
With $A\in\Bbb R^{n\times n}$ as orthogonal and $S(x)$ as sum of
coordinates of $x$, for which $x\in\Bbb R^n$ does $S(Ax \pmod 1) = S(x
\pmod 1)$?
Does the linear combination of eigenvectors of $A$ that correspond to $1$
form part of solution?
Are there any other vectors given easily?