For a completely integrable system, the way of finding first integrals is not formulated in general. A new characterization for integrable systems using the particular tensor field is investigated which is called a recursion operator. A recursion operator $T$ for a vector field $\Delta$ is a diagonizable $(1, 1)$-type tensor field, invariant under $\Delta$ and has vanishing Nijenhuis torsion. One of the important property of $T$ is that $T$ gives constants of the motion (the sequence of first integrals) for the vector field $\Delta$. The purpose of this paper is to discuss a recursion operator $T$ for the geodesic flow on $S^n$.