Computing H-equations with 2-by-2 integral matrices

We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral invertible matrices: “given h1,..., hr; g ¿ PSL2 (Z), decide whether g is algebraic over the subgroup H = <h1,..., hr> <= PSL2 (Z) (i.e., whether there exist a non-trivial H-equation w(x) ¿ H¿<x> such that w(g) = 1) and, in the affirmative case, compute finitely many such H-equations w1(x),..., ws(x) ¿ H¿<x> further satisfying that any w(x) ¿ H¿<x> with w(g) = 1 is a product of conjugates of w1(x),..., ws(x)”. The same problem for square matrices of size 4 and bigger is unsolvable.