New Integrable Hamiltonians with Transcendental Invariants

This paper presents three two-dimensional integrable Hamiltonians, whose second invariant $I_2$ is rational or transcendental in momenta. A third invariant $I_3$ (canonically conjugate to $I_2$) is also found for them, and the motion of the particle explicitly solved. The original Painlevé test does not work very well for these models; however, expansions with four resonances are found in each case.