  We develop an analytical Hamiltonian formalism adapted to the study of the
motion of two planets in co-orbital resonance. The Hamiltonian, averaged over
one of the planetary mean longitude, is expanded in power series of
eccentricities and inclinations. The model, which is valid in the entire
co-orbital region, possesses an integrable approximation modeling the planar
and quasi-circular motions. First, focusing on the fixed points of this
approximation, we highlight relations linking the eigenvectors of the
associated linearized differential system and the existence of certain
remarkable orbits like the elliptic Eulerian Lagrangian configurations, the
Anti-Lagrange (Giuppone et al., 2010) orbits and some second sort orbits
discovered by Poincar\'e. Then, the variational equation is studied in the
vicinity of any quasi-circular periodic solution. The fundamental frequencies
of the trajectory are deduced and possible occurrence of low order resonances
are discussed. Finally, with the help of the construction of a Birkhoff normal
form, we prove that the elliptic Lagrangian equilateral configurations and the
Anti-Lagrange orbits bifurcate from the same fixed point L4.
