We obtain sufficient conditions for the nontrivial solvability of systems of the form and of the corresponding homogeneous systems. It is assumed that the sequences b = ( b, b, b, ...) and λ = ( λ, λ, λ, ...) and the Toeplitz matrix A = ( a) satisfy the conditions . Under these conditions, we construct bounded solutions of homogeneous and inhomogeneous systems of the form indicated above.