Let $S^n\subset\mathbb{R}^{n+1}$ be the standard unit $n$-sphere. The geodesics on $S^n$ are the great circles, which means that the geodesic flow on $TS^n$ can be written down without very much trouble. In this sense, the geodesic flow on $TS^n$ is integrable.
There is another notion of integrablity in classical mechanics, namely Liouville integrability. I would be very surprised if the geodesic flow on $TS^n$ was not Liouville integrable.
My question: Is there a well-known family of $n$ independent Poisson-commuting first integrals for the geodesic flow on $TS^n$?
What I know so far:
I can write down the momentum map $\mu:TS^n\rightarrow\mathfrak{so}^*(n+1)$ corresponding to the natural action of $SO(n+1)$ on $S^n$. This gives $n(n+1)/2$ first integrals, $\mu_i=\left<\mu,\xi_i\right>$, where the $\xi_i$ form a basis for $\mathfrak{so}(n+1)$. However, this does not immediately prove Liouville integrability because these first inegrals do not Poisson commute: $\{\mu_i,\mu_j\}=\left<\mu,[\xi_i,\xi_j]\right>$.