Periods of Negative-regular Continued Fractions. Rational numbers

Periods of pure periodic negative-regular continued fractions
\begin{equation*}\label{mainrepx1} \underbrace{\frac{-1}{b_1}\underset{+}{}\, \frac{-1}{b_2}\underset{+\cdots+}{}\,\frac{-1}{b_n}}_n\underset{+}{}\, \underbrace{\frac{-1}{b_1}\underset{+}{}\, \frac{-1}{b_2}\underset{+\cdots+}{}\,\frac{-1}{b_n}}_n\underset{+\cdots}{}\,, \end{equation*}
where $b_i$ are positive integers, are studied. These continued fractions either converge to irrational numbers, or converge to rational numbers including $0$ and $\infty$, or diverge. Given a rational number $x$ we give a formula for the period of the minimal length representing $x$and prove that it is unique. We also classify the so-called primitive periods. Let $S$ and $ST$ be the standard generators of the modular group $\Gamma$. We prove that any $\mu$ in $\Gamma$ can be represented in the form $ST^{b_1}\cdots ST^{b_n}$, where $\{b_1,\ldots, b_n\}$ is a primitive period. A periodic negative-regular continued fraction diverges essentially if and only if at least one of the three following conditions holds:
$(1)\quad $ $\{b_1,\ldots, b_n\}$ represents the identity;
$(2)\quad$ $\{b_1,\ldots, b_n,b_1,\ldots, b_n\}$ represents the identity;
$(3)\quad $ $\{b_1,\ldots, b_n,b_1,\ldots, b_n,b_1,\ldots, b_n\}$ represents the identity.