Geometrization and integrability

Given a manifold, can one introduce a “good metric” on it, and if yes, in “how many ways”? This is one of the informal versions of the general geometrisation programme, going back to Riemann, Klein and Poincare, but still attracting a substantial interest of geometers. In particular, in dimension 3 the proof by Perelman of Thurston’s geometrisation conjecture became one of the major mathematical events of our time. This example shows also that giving the precise definition of good metric could be the key part of the question.
From the point of view of the theory of integrable systems the question looks quite natural, if one understands a good metric in the sense of the integrability of the corresponding geodesic flow. I will demonstrate the importance of this point of view both for geometry and for the theory of integrable systems.