Let G(D) be a linear partial differential operator on {mathbb {R}}^n, with constant coefficients. Moreover let Omega subset {mathbb {R}}^n be open and Fin L^1_{text {loc}} (Omega , {mathbb {C}}^N). Then any set of the form Af,F:={x∈Ω|(G(D)f)(x)=F(x)},withf∈Wlocg,1(Ω,Ck)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_{f,F}:= \\{ x\\in \\Omega \\, \\vert \\, (G(D)f)(x)=F(x)\\}, \\text { with }f\\in W^{g,1}_{\\text {loc}}(\\Omega , {\\mathbb {C}}^k) \\end{aligned}$$\\end{document}is said to be a G-primitivity domain of F. We provide some results about the structure of G-primitivity domains of F at the points of the (suitably defined) G-nonintegrability set of F. A Lusin type theorem for G(D) is also provided. Finally, we give applications to the Maxwell type system and to the multivariate Cauchy-Riemann system.