N = 2 Integrable Systems and First Integrals Constrained by Scaling Symmetries

Abstract

Integrability of systems in N = 2 dimensions possessing a non-trivial scaling symmetry is reformulated in the context of dynamical symmetries and solvable Lie algebras. A method to construct integrable (Lagrangians) from a dynamical vector field subjected to a scaling symmetry is proposed.

Higher-order superintegrability of a Holt related potential

“…It has been proved the existence, in addition to the original potential V h1 , of two other integrable potentials of Holt type (see e.g. [8] or [16]). They are :…”

“…In addition to the original potential V h1 , the existence has also been proved of two other integrable potentials of Holt type (see e.g. [8] or [16]). They are: (h2) the potential…”

“…In fact, the Holt potentials have been studied by many authors making use of different approaches (Painlevé analysis [4], the direct method [6], Lax equations [8], relation with the Drach potentials [10,30], Lie symmetries [7], scaling symmetries [16]), but in spite of this, we see that they are endowed with certain properties that still remain to be studied. Now, concerning the relation of the superintegrable potential U with the Holt potentials we consider that there are two interesting points to be studied: (i) the existence of higher-dimensional versions of the Holt potentials [26] as a method for obtaining potentials similar to U but with three or more degrees of freedom, and (ii) the duality and coupling-constant metamorphosis between integrable Hamiltonian systems [27,28], and, more specifically, the relation of the Holt potential with the Hénon-Heiles-type Hamiltonians [29] can also lead to the construction of other superintegrable systems.…”

“…Very few systems of this class are known. We mention the Holt [2] and the Fokas-Lagerstrom [3] potentials that are both endowed with a cubic integral (see [4][5][6][7][8][9][10][11][12][13][14][15][16] for some other related papers).…”