  We study arithmetical and combinatorial properties of $\beta$-integers for
$\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq
n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of
$\beta$-fractional positions, which may arise as a result of addition of two
$\beta$-integers. For the infinite word $u_\beta$ coding distances between
consecutive $\beta$-integers, we determine precisely also the balance. The word
$u_\beta$ is the fixed point of the morphism $A \to A^{m-1}B$ and $B\to
A^{m-n-1}B$. In the case $n=1$ the corresponding infinite word $u_\beta$ is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
$n\geq 2$, we illustrate how closely the balance and arithmetical properties of
$\beta$-integers are related.
