Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are linearly independant ?
(recall that in the usual definition of completely integrable flows, one only requires that these are independent almost everywhere or in a dense open set).
Thanks !