We characterised all non-zero vector-fields S ∈ [W^{−1,p}(Ω)]^n , 1 < p < ∞, n ≥ 3, whose potential, φ, linked to S by the elliptic problem ∇·(M ∇φ)= ∇·S, attains a constant value on each of the finitely many connected components of Rn \Ω, where M a symmetric positive definite matrix. Our characterisation states that such S posses a Stokes decomposition and when such S are extended by zero to R^n their Stokes decomposition vanishes identically outside Ω. We also showed that given S ∈ [W^{−1,p} (Ω)]^n there is a unique S_{nm} ∈ [W^{−1,p}(Ω)]^n of minimum norm among all vector-fields that generate the same potential as S on R^n\Ω modulo constants. We showed that when Ω admits the Gauss divergence theorem there is a unique h∗ ∈ W 2,q (Ω) such that S_{nm} = ⟨S, ∇h∗⟩∆q ∇h∗ where q = (p−1)/p and ∆q is the vector q−Laplacian hence each vector-field S ∈ [W^{−1,p}(Ω)]^n can be written as S = ∆v + ∇ψ − ⟨S, ∇h∗⟩∆q ∇h∗ for unique v ∈ [W 1,p (Ω)]n and ψ ∈ Lp (Ω). Finally, we showed that when Ω is Lipschitz, under certain circumstances it is possible to determine S_{nm} from φ on ∂Ω.