We give a complete classification of meromorphically integrable homogeneous potentials V of degree -1 which are real analytic on R-2 {0}. In the more general case when V is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point c with V' (c) = -c, c(1)(2)+ c(2)(2) not equal 0 and Sp(del V-2(c)) subset of {-1, 0, 2}. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case V' (c) = 0.
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