Abstract We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐μ = σ, μ, σ are Radon measures, considering the measure μ(x) = g(x)d x + μus (x̃)(μs (x̄) + d x̄) where x = (x̃,x̄) ∈ ℝ k × ℝ d−k , μus is a uniformly singular measure in x̃ 0 and the measure μs is a singular measure. We proved that for μus -a.e. x̃ 0 the range of the Radon-Nykodim derivative f ~ ( x ~ 0 ) = d μ u s d | μ u s | ( x ~ 0 ) $\begin{array}{} \tilde{f}(\tilde{{\bf x}}_0) = \frac{d \mu_{us}}{d | \mu_{us}|}(\tilde{{\bf x}}_0) \end{array}$ is in the set ∩ ξ̃ ∈P̃ 𝓚er A P̃ ( ξ ) and, if μs is different to zero, for μs -a.e. x̄ 0 the range of the Radon-Nykodim derivative f ¯ ( x ¯ 0 ) = d μ s d | μ s | ( x ¯ 0 ) $\begin{array}{} \bar{f}(\bar{{\bf x}}_0) = \frac{d \mu_{s}}{d | \mu_{s}|}(\bar{{\bf x}}_0) \end{array}$ is in the set ∪ ξ̄ ∈P̄ 𝓚er A P̄ ( ξ ) where P̃ × P̄ = P is a manifold determined by the main symbol A P = A P̃ ⋅ A P̄ of the operator 𝓐.