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In the rst part of the paper we investigate the integrability properties of a second order linear ODE, which is a reduction of the H enon-Heiles systems and a generalization of the Lam e equation. The equation has four regular singular points in the extended complex plane. The investigation of the poly-Painlev e property (the Kruskal test) is done by calculating explicitly the monodromy group of the equation. The result is that there is only one single-valued and necessarily non-analytic rst integral, which is explicitly found using the branching properties of the solutions. In the second part we prove non-integrability of the H enon-Heiles systems in a comprehensive setting. The method used is the Kruskal test and is general. Short title: Integrability properties of some ODEs 34-A20; non-integrability, H enon-Heiles system, holomorphic rst integral, branching

Integrability properties of a generalized Lamé equation: Applications to the Henon–Heiles system