Variational Poisson-Nijenhuis structures for partial differential   equations
  We 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. 
Valentina Golovko, Iosif Krasil'shchik, and Alexander Verbovetsky