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In this paper we present the theorem on Lie integrability by quadratures fortime-independent Hamiltonian systems on symplectic and contact manifolds, andfor time-dependent Hamiltonian systems on cosymplectic and cocontact manifolds.We show that having a solvable Lie algebra of constants of motion for aHamiltonian system is equivalent to having a solvable Lie algebra of symmetriesof the vector field defining the dynamics of the system, which allows us tofind the solutions of the equations of motion by quadratures.

Lie integrability by quadratures for symplectic, cosymplectic, contact and cocontact Hamiltonian systems