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The phase space of an integrable Hamiltonian system is foliated by invarianttori. For an arbitrary Hamiltonian H such a foliation may not exist, but we canartificially construct one through a parameterised family of surfaces, with theintention of finding, in some sense, the closest integrable approximation to H.This is the Poincar\'e inverse problem (PIP). In this paper, we review theavailable methods of solving the PIP and present a new iterative approach whichworks well for the often problematic thin orbits.

Poincar\'e inverse problem and torus construction in phase space