  In this paper we are concerned with completely integrable Hamiltonian systems
in the setting of contact geometry. Unlike the symplectic case, contact
structures are automatically Hamiltonian. Using the Jacobi brackets defined on
contact manifolds, we discuss the commutativity of the first integrals for
contact Hamiltonian systems and introduce the generalized contact action-angle
variables. We exemplify the general scheme in the case of the five-dimensional
toric Sasaki-Einstein spaces $T^{1,1}$ and $Y^{p,q}$.
