Geodesic compatibility and integrability of geodesic flows

We give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a pseudo-Riemannian metric. We prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition. A lot of metrics studed in Riemannian and Kahlerian geometry satisfy such conditions. We apply our results for obtaining an infinite family (hierarchy) of completely integrable flows on the complex projective plane CPn. (C) 2003 American Institute of Physics.