This paper deals with a normal form of Hamiltonian 1:1:3 resonance. It is not integrable, and we write it using the basic invariants. Also, we identify the coefficients of the terms that remain in the normalization procedure. Then, by choosing different potential functions, we consider three integrable subfamilies of the Hamiltonian with a discrete symmetry. They are containing a Hamiltonian in a 3D Greene case, a generalized Hénon-Heiles Hamiltonian, and a quartic Hamiltonian. We consider the detuning parameters and analyze the bifurcations.