Coincidence of invariant measure for the alternate base transformations

We characterize all pairs $(\beta ,n),({\beta }^{\mathrm{\prime }},m)$ such that the alternate $(\beta ,n)$ and $({\beta }^{\mathrm{\prime }},m)$-transformations $K_{(\beta ,n)}$ and $K_{({\beta }^{\mathrm{\prime }},m)}$ have the same absolutely continuous invariant measure, where $K_{(\beta ,n)}(i,x)=(i+1\mathrm{m}\mathrm{o}\mathrm{d}2,T_i(x))$ with $i\in \{0,1\}$, $T_0(x)=T_{\beta }(x)=\beta x\mathrm{m}\mathrm{o}\mathrm{d}1$, $T_1(x)=T_n(x)=nx\mathrm{m}\mathrm{o}\mathrm{d}1$ with $\beta >1$ real and $n\ge 2$ an integer.