On Hamiltonian systems in two degrees of freedom with invariants quartic in the momenta of form p21p22⋅⋅⋅

Abstract

A search is made for autonomous Hamiltonian systems in two degrees of freedom which admit a second invariant quartic in the momenta with leading term p21p22/ 2. A sufficient condition for the resulting functional equation to possess solutions is deduced and a family of integrable systems is identified, which under the equivalence class of linear transformations reduce to a simpler integrable system found originally by Bozis. The method of Lax pairs is used to find further solutions to the functional equation and give new classes of integrable but nonseparable Hamiltonians.