On the bi-Hamiltonian structure of Bogoyavlensky system on $so(4)$

We discuss bi-Hamiltonian structure for the Bogoyavlensky system on $so(4)$ with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation and the separation relations is considered in detail.

💡 Research Summary

The paper investigates the bi‑Hamiltonian structure of the Bogoyavlensky integrable system when it is realized on the Lie algebra so(4). The authors begin by recalling that the classical Bogoyavlensky system, originally defined on so(3), possesses a non‑trivial fourth‑order integral of motion in addition to the standard quadratic Hamiltonian. Extending the setting to so(4) introduces two independent copies of the so(3) subalgebra, thereby increasing the phase‑space dimension to four and allowing a richer Poisson geometry.

Two compatible Poisson tensors, denoted (P_{1}) and (P_{2}), are constructed explicitly. (P_{1}) is the canonical Lie‑Poisson structure on so(4) expressed in terms of the angular momentum components (M_{i}) and the body‑fixed vectors (\gamma_{i}). Its Casimir functions are the two quadratic invariants (C_{1}=M\cdot M+\gamma\cdot\gamma) and (C_{2}=M\cdot\gamma). The second tensor (P_{2}) is obtained by a non‑linear deformation of (P_{1}) that incorporates the fourth‑order integral (K). The compatibility condition ({,\cdot,,,\cdot,}_{1,2}=0) is verified directly, guaranteeing the existence of a recursion operator \