A new characterization for integrable systems using the particular tensor field is investigated which is called a recursion operator. In this paper, we discuss recursion operators and construct the one for the Kepler dynamics by means of the Runge-Lenz-Pauli vector. We also give an explicit construction of recursion operators for the geodesic flows on the upper half-space, then-dimensional sphere with canonical metrics and the Minkowski space.

Hamiltonian systems, integrable systems, recursion operator, geodesic flows.