We are concerned with completely integrable Hamiltonian systems and generalized action-angle coordinates in the setting of contact geometry. We investigate the deformations of the Sasaki-Einstein structures, keeping the Reeb vector field fixed, but changing the contact form. We examine the modifications of the action-angle coordinates by the Sasaki-Ricci flow. We then pass to the particular cases of the contact structures of the five-dimensional Sasaki-Einstein manifolds $T^{1,1}$ and $Y^{p,q}$.