  We study natural lifting operations from a bundle E over R to the dual bundle
of its first-jet bundle. The main purpose is to define a complete lift of a
type (1,1) tensor field on E and to understand all features of its
construction. Various other lifting operations of tensorial objects on E are
needed for that purpose. We prove that the complete lift of a type (1,1) tensor
with vanishing Nijenhuis torsion gives rise to a Poisson-Nijenhuis structure on
the dual of the first-jet bundle, and discuss in detail how the construction of
associated Darboux-Nijenhuis coordinates can be carried out.
