On the generalization of classical Zernike system

Abstract

We generalize the results obtained recently (Blasco et al 2023 Nonlinearity 36 1143) by providing a very simple proof of the superintegrability of the Hamiltonian H = p ⃗ 2 + F ( q ⃗ ⋅ p ⃗ ) , q ⃗ , p ⃗ ∈ R 2 , for any analytic function F. The additional integral of motion is constructed explicitly and shown to reduce to a polynomial in canonical variables for polynomial F. The generalization to the case q ⃗ , p ⃗ ∈ R n is sketched.