NL bialgebras

In this paper, we introduce the concept of (weak) NL bialgebras. These structures consist of a Lie bialgebra $(\text{g},[\cdot ,\cdot ],\delta )$ equipped with a Nijenhuis structure on the Lie algebra $(\text{g},[\cdot ,\cdot ])$, satisfying specific compatibility conditions. This construction is analogous to Poisson-Nijenhuis structures studied in the context of integrable systems. We further investigate NL bialgebras that generate a compatible hierarchy of bialgebras, both on the original Lie algebra and its deformed versions, through the Nijenhuis structure of any order. Additionally, we demonstrate that the underlying algebraic structure of a particular case of the Euler-top system is a weak NL bialgebra.