Integrability of Hamiltonian systems and transseries expansions

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This paper studies analytic Liouville-nonintegrable and C ∞-Liouville-integrable Hamiltonian systems with two degrees of freedom. We prove the property for a class of Hamiltonians more general than the one studied in Gorni and Zampieri (Differ Geom Appl 22:287–296, 2005). We also show that a certain monodromy property of an ordinary differential equation obtained as a subsystem of a given Hamiltonian and the transseries expansion of a first integral play an important role in the analysis (cf. (4)). In the former half the analytic Liouville-nonintegrability for a class of Hamiltonians satisfying the above condition is shown. For these analytic nonintegrable Hamiltonians one cannot construct nonanalytic first integrals concretely as in Gorni and Zampieri (Differ Geom Appl 22:287–296, 2005). In the latter half, the nonanalytic integrability from the viewpoint of a transseries expansion of a first integral is discussed. More precisely, we construct a first integral in a formal transseries expansion in a general situation. Then we show convergence of transseries or existence of the first integral being asymptotically equal to a given formal transseries solution.