On rational integrability of Euler equations on Lie algebra so(4, C), revisited

Sasho Ivanov Popov (1) , Witold Respondek (2) , Jean-Marie Strelcyn (3)

1 Institute of Metal Science, Bulgarian Academy of Sciences
2 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

We consider the Euler equations on the Lie algebra so(4,C) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones (so called fourth integral), then it has a polynomial fourth first integral. This is a consequence of a more general fact that for these systems the existence of a Darboux polynomial with non vanishing cofactor implies the existence of a polynomial fourth integral.

Mots clés: Euler equations; Rational and polynomial first integrals; Darboux polynomials

Domaines: Systèmes dynamiques [math.DS]

Sasho Ivanov Popov, Witold Respondek, Jean-Marie Strelcyn. On rational integrability of Euler equations on Lie algebra so(4, C), revisited. 2009. ⟨hal-00372958⟩