  In this paper we explore the general conditions in order that a 2-dimensional
natural Hamiltonian system possess a second invariant which is a polynomial in
the momenta and is therefore Liouville integrable. We examine the possibility
that the invariant is preserved by the Hamiltonian flow on a given energy
hypersurface only (weak integrability) and derive the additional requirement
necessary to have conservation at arbitrary energy (strong integrability).
Using null complex coordinates, we show that the leading order coefficient of
the polynomial is an arbitrary holomorphic function in the case of weak
integrability and a polynomial in the coordinates in the strongly integrable
one. We review the results obtained so far with strong invariants up to degree
four and provide some new examples of weakly integrable systems with linear and
quadratic invariants.
