We briefly describe the construction of Stäkel–Killing and Killing–Yano tensors on toric Sasaki–Einstein manifolds without working out intricate generalized Killing equations. The integrals of geodesic motions are expressed in terms of Killing vectors and Killing–Yano tensors of the homogeneous Sasaki–Einstein space $$T^{1,1}$$ . We discuss the integrability of geodesics and construct explicitly the action-angle variables. Two pairs of frequencies of the geodesic motions are resonant giving way to chaotic behavior when the system is perturbed.