Quasiperiodic and mixed commutator factorizations in free products of groups

It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation [x1, y1] . . . [x_k, y_k] = zⁿ, where n ≥ 2k, in the free product F of groups without nontrivial elements of order ≤ n implies that z is conjugate to an element of a free factor of F. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.