  In this paper, we study Lie bialgebra analogs of Poisson-Nijenhuis structures
and introduce the notion of (weak)Nijenhuis-Lie bialgebra, a Lie bialgebra with
a Nijenhuis structure on a Lie algebra satisfying compatibility conditions. We
show that a Nijenhuis-Lie bialgebra gives rise to a hierarchy of compatible Lie
bialgebras on the initial Lie algebra and deformed Lie algebras by means of the
Nijenhuis structure of any order. We show that the underlying algebraic
structure of Euler-top system is a weak Nijenhuis-Lie bialgebra.
