Explicit integrability for Hamiltonian systems and the Painlevé conjecture

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Abstract
We analyze a class of Hamiltonian systems in two dimensions for which we proved that a second constant of motion exists. It is shown that, using the two first integrals, the equations of motion can be written in a form which allows their integration by quadratures. An analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex‐time plane. It is shown explicitly that the systems belonging to this class possess the ‘‘weak Painlevé’’ property, i.e., their solutions in complex time can present singularities of a specific algebraic type.