A Note on Integrable Mechanical Systems on Surfaces

Let S be a compact, connected surface and H in C^2(T^* S) a Tonelli Hamiltonian. This note extends V. V. Kozlov's result on the Euler characteristic of S when H is real-analytically integrable, using a definition of topologically-tame integrability called semisimplicity. Theorem: If H is 2-semisimple, then S has non-negative Euler characteristic; if H is 1-semisimple and reversible, then S has positive Euler characteristic.