Integrability versus topology of configuration manifolds and domains of   possible motions
  We establish a generic sufficient condition for a compact $n$-dimensional manifold bearing an integrable geodesic flow to be the $n$-torus. As a complementary result, we show that in the case of domains of possible motions with boundary, the first Betti number of the domain of possible motions may be arbitrarily large. 
M. Rudnev, V. Ten