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We prove that the only meromorphically integrable planar homogeneouspotential of degree k <> -2,0,2 having a multiple Darboux point is thepotential invariant by rotation. This case is a singular case of theMaciejewski-Przybylska relation on eigenvalues at Darboux points of homogeneouspotentials, and needed before a case by case special analysis. The moststriking application of this Theorem is the complete classification ofintegrable real analytic homogeneous potentials in the plane of negativedegree.

Meromorphically integrable homogeneous potentials with multiple Darboux points