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We develop an analytical Hamiltonian formalism adapted to the study of themotion of two planets in co-orbital resonance. The Hamiltonian, averaged overone of the planetary mean longitude, is expanded in power series ofeccentricities and inclinations. The model, which is valid in the entireco-orbital region, possesses an integrable approximation modeling the planarand quasi-circular motions. First, focusing on the fixed points of thisapproximation, we highlight relations linking the eigenvectors of theassociated linearized differential system and the existence of certainremarkable orbits like the elliptic Eulerian Lagrangian configurations, theAnti-Lagrange (Giuppone et al., 2010) orbits and some second sort orbitsdiscovered by Poincar\'e. Then, the variational equation is studied in thevicinity of any quasi-circular periodic solution. The fundamental frequenciesof the trajectory are deduced and possible occurrence of low order resonancesare discussed. Finally, with the help of the construction of a Birkhoff normalform, we prove that the elliptic Lagrangian equilateral configurations and theAnti-Lagrange orbits bifurcate from the same fixed point L4.

On the co-orbital motion of two planets in quasi-circular orbits