  We propose a new condition $\aleph$ which enables to get new results on
integrable geodesic flows on closed surfaces. This paper has two parts. In the
first, we strengthen Kozlov's theorem on non-integrability on surfaces of
higher genus. In the second, we study integrable geodesic flows on 2-torus. Our
main result for 2-torus describes the phase portraits of integrable flows. We
prove that they are essentially standard outside, what we call, separatrix
chains. The complement to the union of the separatrix chains is $C^0$-foliated
by invariant sections of the bundle.
