Kepler dynamics on a α-deformed Poisson manifold: Recursion operators and master symmetries

The problem of Kepler dynamics on a [Formula: see text]-deformed Poisson manifold is addressed. The Hamiltonian function is defined and the related Hamiltonian vector field governing the dynamics is derived, which leads to a modified Newton second law. [Formula: see text]-deformed momentum and Laplace–Runge–Lenz vectors are considered, generating [Formula: see text] and [Formula: see text] dynamical symmetry groups. The corresponding first Casimir operators of [Formula: see text] and [Formula: see text] are, respectively, obtained. The recursion operators are constructed and used to compute the integrals of motion in action-angle coordinates. Main relevant properties such as quasi-bi-Hamiltonian systems, bi-Hamiltonian systems, and master symmetries are deducted. Then, a plethora of conserved quantities is highlighted.