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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(〈 x, y ; 〉).

The two variable substitution problem for free products of groups