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

TL;DR

This paper critiques recent claims of novelty in integrable Hamiltonian systems with quartic and cubic integrals, showing they are re-discoveries of earlier known systems by the author and others.

Contribution

The paper clarifies that two recently published systems are not new but are special cases of previously known integrable systems from earlier works.

Findings

The systems in Valent's first paper are special cases of systems from 2006.

The systems in Valent's second paper are not new and are related to systems from 2002 and even earlier.

Some systems claimed as original are actually rediscoveries of previously published results.

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.