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We study the planar and the three-dimensional 1:2 resonant motion with Neptune (N1:2) in the framework of the elliptic restricted three-body problem. We examine the dynamics at the 1:2 mean motion resonance with Neptune in a hierarchy of models: the planar circular, the planar elliptic, the three-dimensional circular and the three-dimensional elliptic restricted three body problem. We start from the planar circular model in which we compute families of symmetric periodic orbits with stable and unstable segments. Using the circular planar model as the basic model, we proceed to more realistic models, namely the planar elliptic and the 3D circular problem. Families of symmetric periodic orbits of the above models bifurcate from the families of periodic orbits of the planar circular problem. Moreover, we compute families of symmetric periodic orbits in the 3D elliptic restricted three-body problem. These new families bifurcate from the families of periodic orbits of the planar elliptic and the 3D circular problem. The stability of all orbits is studied and the structure of the phase space is discussed in detail.