An analytical solution to the Hill problem Hamiltonian expanded about the
libration points has been obtained by means of perturbation techniques. In
order to compute the higher orders of the perturbation solution that are needed
to capture all the relevant periodic orbits originated from the libration
points within a reasonable accuracy, the normalization is approached in complex
variables. The validity of the solution extends to energy values considerably
far away from that of the libration points and, therefore, can be used in the
computation of Halo orbits as an alternative to the classical
Lindstedt-Poincar\'e approach. Furthermore, the theory correctly predicts the
existence of the two-lane bridge of periodic orbits linking the families of
planar and vertical Lyapunov orbits.Comment: 28 pages, 8 figure
