2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/218647We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.12 pages; v2: the content isn't changed, submission metadata fixed; v3: a minor correctionDifferential GeometryMathematical PhysicsExactly Solvable and Integrable Systems37K05; 35Q53text