Commutators and powers of infinite unitriangular matrices

For a given infinite unitriangular matrix A we discuss the matrix equations xk=A, ωkm1,...,mk(x1,...,xk)=A and ekl,m1,...,mk(x,y)=A in variables x,x1,...,xk,y. As a main result, we provide the necessary and sufficient conditions for solvability of these equations.