We study the dynamics near the origin of a family of autonomous Hamiltonian systems associated with the polynomial function H=1/2(x(2)+X-2)+1/2(y(2)+Y-2)+delta(z(2)+Z(2)) +[alpha(x(4)+y(4)+z(4))+beta(x(2)y(2)+x(2)z(2)+y(2)z(2))]+H-& lowast;, where delta=+/- 1, that is, in 1:1:2 or 1:1: -2 resonance and depending on alpha, beta real parameters, and H-& lowast; is any polynomial function of degree greater than four. The flow of the Hamiltonian vector field is reconstructed using normal forms and applying singular reduction theory, obtaining the reduced Hamiltonian defined on the orbit space. From critical points of the reduced Hamiltonian, we prove the existence of periodic solutions together with their linear stability using the averaging theory for Hamiltonian systems. In fact, in the case of resonance 1:1:2 there are at most seven families of periodic solutions for every energy level h>0, while in the case 1:1: -2 there are at most six families of periodic solutions for every energy level h>0 and one family for every h<0. Moreover, the bifurcations of periodic solutions are characterized in terms of the parameters. Also, we determine KAM 3-tori encasing the linearly stable periodic solutions. For the integrable case ( beta=0), we can apply our analysis. In fact, we get three periodic solutions for delta=1, two periodic solutions for delta=-1 and KAM tori. During the work, we highlight the important differences intrinsic to the resonances 1:1:delta and 1:1: 2 delta on the reduced space with the appropriated symplectic coordinates.