Comments on two papers by Galliano Valent, concerning integrable Hamiltonian systems admitting quartic and cubic integrals

Abstract

In this note we comment on two recently published papers by G. Valent: The 1st is the preprint "On a Class of Integrable Systems with a quartic First Integral, arXiv:1304.5859. April 22, (2013)". We show that the two integrable Hamiltonian systems introduced in this reprint as original results are not new. They are special cases of two systems introduced by the present author in 2006 in two papers [6] and [5]. The second paper is "On a Class of Integrable Systems with a Cubic First Integral, Commun. Math. Phys. 299, 631{649 (2010), In that paper two integrable Hamiltonian systems admitting a cubic integral were introduced. Those systems were referred to as original results by Tsiganov in [12], Vershilov and Tsiganov in [13], Bialy and Mironov in [15] and by Gibbons et al in [14]. We show that those systems are not new. Both can be obtained as special cases of one system introduced by us in [4] (2002) and one of them is a special case of a much earlier version [1] published 24 years earlier.