Completely and partially integrable systems of total differential equations

Constrained Hamiltonian systems are investigated by using the Hamilton-Jacobi method. Integration of a set of equations of motion and the action function is discussed. It is shown that we have two types of integrable systems: a) Partially integrable systems, where the set of equations of motion is only integrable. b) Completely integrable systems, where the set of equations of motion and the action function is integrable. Two examples are studied.