The Post–Winternitz system on spherical and hyperbolic spaces: A proof of the superintegrability making use of complex functions and a curvature-dependent formalism

Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

“…In any case it is natural to ask what are the relations between the two structures. Second, the complex functions method presented in this paper (as well in some other previous papers mentioned in the Introduction) is restricted to the two dimensional case; it is convenient to study the generalization to the three-dimensional case (the multiple separability of threedimensional systems was first studied in [17]) and also to constant curvature spaces (the superintegrability of some particular systems was studied in [14,21,39,36] making use of curvature-dependent coordinates); the generalization of the system studied in this paper must be done making use of curvature-dependent parabolic coordinates. Third, the complex functions (A, B) or (M a , M b ) are important for two reasons since they determine the integrals of motion (AB * or M a M * b ) and also the geometric structures; probably there are some additional properties hidden behind these functions deserving to be studied making use of tools of complex differential geometry.…”

“…The important point is that this property (superintegrability related with the existence of some complex functions satisfying certain Poisson brackets properties) is not just an exclusive characteristic of the harmonic oscillator H mn . In fact, it has been recently proved that other superintegrable systems also admit a complex factorization for the additional constants of motion (as the above mentioned SW nonlinear isotonic oscillator [34], Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems [37] and also some particular systems defined in spaces with constant curvature [39,36]).…”

Curvature as an integrable deformation

“…If we take m = 2 and n = 1, then ω x = 2ω y = 2ω, ξ = 2x and p ξ = p x /2. The Hamiltonian (26) and the integrals (32) turn out to be…”

“…we will say that we have obtained a superintegrable curved generalization of the Euclidean superintegrable Hamiltonian H. The explicit curvature-dependent description of S 2 and H 2 is well-known in the literature and can be found, for instance, in [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] (see also references therein) where it has been mainly considered in the classification and description of superintegrable systems on these two spaces. In this contribution we will present several recent works in which this geometric framework has been applied for non-superintegrable systems where the lack of additional symmetries forces to make use of a purely integrable perturbation approach.…”

Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

“…The important point is that this property (superintegrability related with the existence of some complex functions satisfying certain Poisson brackets properties) is not just an exclusive characteristic of the harmonic oscillator H mn . In fact, it has been recently proved that other superintegrable systems also admit a complex factorization for the additional constants of motion (as the above mentioned SW nonlinear isotonic oscillator [2], Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems [3] and also some particular systems defined in spaces with constant curvature [13,14]).…”