On rational integrability of Euler equations on Lie algebra , revisited

• Published in: Physics letters. A Vol. 373; no. 29; pp. 2445 - 2453
• Main Authors: Popov, S.I., Respondek, W., Strelcyn, J.-M.
• Format: Journal Article
• Language: English
• Published: 01.06.2009
• ISSN: 0375-9601
• DOI: 10.1016/j.physleta.2009.04.075

We consider the Euler equations on the Lie algebra [MathML equation] 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.