Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations

Given a Lie-Poisson completely integrable bi-Hamiltonian system, in the new paper arXiv:1609.07438 we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures that underly the dynamics of the deformed system and by making use of the non-abelian group law, one may obtain two completely integrable Hamiltonian systems on the direct product of the non-abelian group by itself. By construction, both systems admit reduction, via the multiplication in the non-abelian group, to the initial deformed bi-Hamiltonian system. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.

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