Let (N,α,G) be a covariant system and let (π, U) be a covariant representation of (N,α,G) on a Hilbert space H. In this note, we investigate the representation of the covariance algebra M and the σ-weakly closed subalgebra A generated by π(N) and {Ug}g≥0 in the case of G = Z or R when there exists a pure, full, A-invariant subspace of H.