  A recursion operator for a geodesic flow, in a noncommutative (NC) phase
space endowed with a Minkowski metric, is constructed and discussed in this
work. A NC Hamiltonian function $\mathcal{H}_{nc}$ describing the dynamics of a
free particle system in such a phase space, equipped with a noncommutative
symplectic form $\omega_{nc}$ is defined. A related NC Poisson bracket is
obtained. This permits to construct the NC Hamiltonian vector field, also
called NC geodesic flow. Further, using a canonical transformation induced by a
generating function from the Hamilton-Jacobi equation, we obtain a relationship
between old and new coordinates, and their conjugate momenta. These new
coordinates are used to re-write the NC recursion operator in a simpler form,
and to deduce the corresponding constants of motion. Finally, all obtained
physical quantities are re-expressed and analyzed in the initial NC canonical
coordinates.
