The algebra \({\mathcal R}\) generated by an abstract singular operator S, Carleman shift and continuous coefficients is investigated. For the algebra \({\mathcal R}\) the symbol algebra Sym \({\mathcal R}\) has been constructed. The Noetherian criterion for operators from \({\mathcal R}\) is given in symbol terms. In contrast to a well-known case (S is a one- dimensional singular integral operator) the symbol algebra Sym \({\mathcal R}\) here is a \(C^*\)-algebra with irreducible representations of one-, two- or four-dimensions. Some concrete examples have been considered.