  Methods of Hamiltonian dynamics are applied to study the geodesic flow on the
resolved conifolds over Sasaki-Einstein space $T^{1,1}$. We construct
explicitly the constants of motion and prove complete integrability of
geodesics in the five-dimensional Sasaki-Einstein space $T^{1,1}$ and its
Calabi-Yau metric cone. The singularity at the apex of the metric cone can be
smoothed out in two different ways. Using the small resolution the geodesic
motion on the resolved conifold remains completely integrable. Instead, in the
case of the deformation of the conifold the complete integrability is lost.
