Singularity of the distribution of a random variable represented by an $A_2$-continued fraction with independent elements

We study the properties of the distribution of the random variable \[ \xi =\frac {1}{\eta _1+\frac {1}{\eta _2+\cdots }}, \] where $\eta _k$ are independent random variables such that $\mathsf {P}\{\eta _k=\alpha _1\}=p_{\alpha _1k}\geq 0$, $\mathsf {P}\{\eta _k=\alpha _2\}=p_{\alpha _2k}\geq 0$, $0<\alpha _1<\alpha _2$, $\alpha _1\alpha _2\geq \frac {1}{2}$, $p_{\alpha _1k}+p_{\alpha _2k}=1$. It is proved that the distribution of $\xi$ cannot be absolutely continuous. We find the criteria for the distribution of $\xi$ to belong to one of the two types of singular distributions, Cantor and Salem types, depending on topological and metric properties of the topological support of the distribution.