  This paper illustrates the application of Lie transform normal-form theory to
the construction of the 1:2 resonant normal form corresponding to a wide class
of natural Hamiltonian systems. We show how to compute the bifurcations of the
main periodic orbits in a potential with double reflection symmetries. The
stability analysis of the normal modes and of the periodic orbits in general
position allows us to get overall informations on the phase-space structure of
systems in which this resonance is dominating. As an example we apply these
results to a class of models useful as galactic potentials.
