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A detailed account is given of a semi-numerical perturbation method which has been proposed and improved upon in a succession of previous papers. The method analyses the first order effect of a small perturbation applied to a non-trivial two-degreeof-freedom separable Hamiltonian system (including the description of resonances) and construct approximate surfaces of section of the perturbed system. When the separable Hamiltonian system is already the description of a resonance, as it is the case in the problems we have investigated in the previous papers, these resonances are actually secondary resonances. The key role of the method is the numerical description of the angle-action variables of the separable system. The method is thus able to describe the perturbations of non-trivial separable systems and is not confined to the analysis of a small neigborhood of their periodic orbits.