Isoenergetic classification of integrable Hamiltonian systems in a neighborhood of a simple elliptic point (Q1898530)

The authors consider a smooth \(2n\)-dimensional symplectic manifold \(M\) and a smooth Hamiltonian function \(H\) and denote by \(X_1\) a Hamiltonian vector field with the Hamiltonian which is called integrable Hamiltonian vector field in a domain \(D \subseteq M\) if there exist \(C^\infty\)-smooth functions \(H_2, \dots, H_n\) such that the collection of the functions is involutive and independent on an open dense subset. Then take \(p\) which is an isolated singular point and, therefore, a singular point of every Hamiltonian vector field. And the authors intend to classify the integrable Hamiltonian vector field in some especially constructed invariant neighbourhood around \(p\) which is simple elliptic. And the authors prove a theorem saying that the integrable Hamiltonian vector fields \(X\), \(X'\) are isoenergetic equivalent in some neighbourhoods \(V\), \(V'\) of their simple elliptic singular points \(p\), \(p'\) if and only if the indices of the points \(p\) and \(p'\) are the same.