In this talk, I will introduce the notion of a Nijenhuis-Lie
bialgebra as a Nijenhuis endomorphism $n: {\frak g} \to {\frak g}$ on a
Lie algebra ${\frak g}$ which is compatible, in a suitable sense, with a
Lie bialgebra structure on ${\frak g}$. An interesting example (the
Euler top) that motivates the previous definition and some results on
the algebraic structure of a Nijenhuis-Lie bialgebra will be presented.
I will also consider the Nijenhuis-Lie bialgebra in the case that Lie
bialgebras are coboundary which turns to the $r$-$n$ structures. The
Nijenhuis-Lie bialgebra structures are a starting point to get a deeper
insight into the underlying geometric structures of the bi-Hamiltonian
systems on Poisson-Lie groups.