A general scheme to obtain finite-dimensional integrable Hamiltonian systems by applying the so-called nonlinearization procedure to the \(n\times n\) matrix spectral problem is briefly described. This general scheme is illustrated in detail by the example of \(3\times 3\) AKNS matrix spectral problem and its adjoint spectral problem associated with the three wave interaction equations. The characteristic polynomial of the solution matrix of the stationary zero-curvature equation is used to generate an involutive system of conserved integrals of the obtained finite-dimensional Hamiltonian system. Two generators of involutive systems are introduced, from which the functional independence of conserved integrals is rigorously proved and, finally, the complete integrability of the obtained Hamiltonian system in the Liouville sense is verified.