  Let $R$ be the ring of $S$-integers in a number field $K$. Let
$\mathcal{B}=\{\beta, \beta^{\ast}\}$ be the multi-set of roots of a nonzero
quadratic polynomial over $R$. There are varieties $V(\mathcal{B})_{N,k}$
defined over $R$ parametrizing periodic continued fractions $[b_1,\ldots ,
b_N,\overline{a_1,\ldots ,a_k}]$ for $\beta$ or $\beta^{\ast}$. We study the
$R$-points on these varieties, finding contrasting behavior according to
whether groups of units are infinite or not. If $R$ is the rational integers or
the ring of integers in an imaginary quadratic field, we prove that the
$R$-points of $V(\mathcal{B})_{N,k}$ are not Zariski dense. On the other hand,
suppose that $\beta\not\in K\cup\{\infty\}$, $R^\times$ is infinite, and that
there are infinitely many units in the (left) order $R_\beta$ of $\beta
R+R\subseteq K(\beta)$ with norm to $K$ equal to $(-1)^k$. Then we prove that
the $R$-points on $V(\mathcal{B})_{1,k}$ are Zariski dense for $k\geq 8$ and
the $R$-points on $V(\mathcal{B})_{0,k}$ are Zariski dense for $k\geq 9$. We
also prove that $V(\mathcal{B})_{1,k}$ and $V(\mathcal{B})_{0,k}$ are
$K$-rational irreducible varieties for $k$ sufficiently large.
