Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

Abstract

In the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

More on superintegrable models on spaces of constant curvature

“…Finally, we would like to remark that our approach can be directly used to constract superintegrable models of spaces of constant curvature and separating in other than geodesic polar coordinates. In principle it can also be applied to study superintegrability of 2D integrable systems with a noncontant curvature presented for instance in [33] Appendix A A convenient way to compute the action J ϕ (L) given in ( 6) is to calculate first its partial derivative…”