Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points

Abstract

The aim of this paper is to describe the structure of an integrable Hamiltonian vector field X _H on an invariant subset V of four-dimensional C ^∞-smooth symplectic manifold M, the subset V containing a singular point p of X _H together with all of its orbits for which p is the limit set. Let H : M → ℝ be a C ^∞-smooth function on M (Hamiltonian) and K be an additional (smooth) integral of the field X _H . The pair (X _H , K) is called an integrable Hamiltonian vector field (briefly, IHVF) if and only if functions H, K are independent in some open dense subset of M (or in a region under consideration). Let p be a singular point of X _H . Without loss of generality we assume H(p) = K(p) = 0. Henceforth we consider eigenvalues of p to be simple.