I am interested in the following question: Given an $n$-tuple of matrices $(A_1, \dots, A_n)\in SL(2,\mathbb R)^n$, does there exist a matrix $B\in SL(2,\mathbb R)$ such that $BA_jB^{-1}\in SL(2,\mathbb Z)$ for any $j$?

If $n=1$, i.e., in the case of a single matrix $A\in SL(2,\mathbb R)$, it is quite easy to see that $A$ is conjugate to a matrix with integer entries if and only if $tr A\in \mathbb Z$. So, the interesting case is really $n\ge 2$ and I would like to have a characterization of $n$-tuple of matrices simultaneously conjugate to ones in $SL(2,\mathbb Z)$. Preferably, the criterion should be easy to check (of course, if there is one.)