GENERALIZED SEPARABILITY FOR A HAMILTONIAN WITH NONSEPARABLE QUARTIC POTENTIAL

We examine a family of two-dimensional Hamiltonians with quartic potentials which are invariant under reflection of each variable separately. Five integrable cases are known, four of which are separable in Cartesian, polar or parabolic coordinates. For the remaining case the separation variables cannot be obtained through a simple transformation of coordinates but involve both positions and momenta. We derive these separation variables with the help of the singular (Painleve) expansions of the solutions of the equations of motion and give explicitly the Lax pair for the associated Hamiltonian.