We consider average wave operators constructed for singular unitary operators U ₁ and U ₂ and a bounded identification operator A. In the case of rank-two commutator AU ₁-U ₂A, we show that averaged wave operators of past and future exist or do not exist simultaneously, and if they exist, then they must coincide. As consequence, we obtain some results concerning the boundary behavior of Cauchy type integrals. Bibliography: 9 titles.