On small bases for which $1$ has countably many expansions

Let $q\in (1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $({\delta }_i{)}_{i=1}^{\mathrm{\infty }}\in \{0,1{\}}^{\mathbb{N}}$ satisfying $$x=\sum _{i=1}^{\mathrm{\infty }}\frac{{\delta }_i}{q^i}\mathrm{.}$$ Let ${\mathcal{B}}_{{\mathrm{\aleph }}_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let ${\mathcal{B}}_{1,{\mathrm{\aleph }}_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \cite{Sidorov6} it was shown that $\min {\mathcal{B}}_{{\mathrm{\aleph }}_0}=\min {\mathcal{B}}_{1,{\mathrm{\aleph }}_0}=\frac{1+\sqrt{5}}{2},$ and in \cite{Baker} it was shown that ${\mathcal{B}}_{{\mathrm{\aleph }}_0}\cap (\frac{1+\sqrt{5}}{2},q_1]=\{q_1\}$, where $q_1(\approx 1.64541)$ is the positive root of $x^6-x^4-x^3-2x^2-x-1=0$. In this paper we show that the second smallest point of ${\mathcal{B}}_{1,{\mathrm{\aleph }}_0}$ is $q_3(\approx 1.68042)$, the positive root of $x^5-x^4-x^3-x+1=0$. Enroute to proving this result we show that $$\mathcal{B}$_{\aleph_0}\cap(q_1, q_3]={ q_2, q_3}$,where$q_2(\approx1.65462)$isthepositiverootof$x^6-2x^4-x^3-1=0$.