  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.
