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Let $V\in\mathbb{Q}(i)(\a_1,\dots,\a_n)(\q_1,\q_2)$ be a rationallyparametrized planar homogeneous potential of homogeneity degree $k\neq -2, 0,2$. We design an algorithm that computes polynomial \emph{necessary} conditionson the parameters $(\a_1,\dots,\a_n)$ such that the dynamical system associatedto the potential $V$ is integrable. These conditions originate from those ofthe Morales-Ramis-Sim\'o integrability criterion near all Darboux points. Theimplementation of the algorithm allows to treat applications that were out ofreach before, for instance concerning the non-integrability of polynomialpotentials up to degree $9$. Another striking application is the first completeproof of the non-integrability of the \emph{collinear three body problem}.

Computing necessary integrability conditions for planar parametrized homogeneous potentials