Cyclic palindromes and periodic continued fractions

The talk is based on the results of a joint paper with I.A. Tlyustangelov.
Since the times of Lagrange it has been known that for each rational $r>1$ different from a perfect square we have the following expansion into continued fraction:
$$ \sqrt{r}\,=\,\bigl[a_0;\overline{a_{1},a_{2},\ldots,a_{2},a_{1},2a_{0}\mathstrut}\,\bigr]. $$
Particularly, a period of this continued fraction read back to front is again a period. We call such periods cyclic palindromic and prove the following criterion.
Theorem. The continued fraction of a quadratic irrationality $\alpha$ has a cyclic palindromic period if, and only if one of the following statements holds:
$(1)$ $\alpha\sim\beta:\ \beta^2\in\mathbb{Q}$;
$(2)$ $\alpha\sim\beta:\ (\beta-1/2)^2\in\mathbb{Q}$;
$(3)$ $\alpha\sim\beta:\ \beta\bar\beta=1$;
$(4)$ $\alpha\sim\beta:\ \beta\bar\beta=-1$.
Moreover, $(2)$ is equivalent to $(3)$.