The two variable substitution problem for free products of groups

We consider equations of the form W(x,y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x,y) = U has only finitely many solutions, up to the action of the stabilizer of W(x,y) in Aut().