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

Generalized action-angle coordinates in toric contact spaces