  We give a complete classification of meromorphically integrable homogeneous
potentials $V$ of degree $-1$ which are real analytic on $\mathbb{R}^2\setminus
\{0\}$. 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 $V'(c)=-c,\; c_1^2+c_2^2\neq 0$ and
$\hbox{Sp}(\nabla^2 V(c)) \subset\{-1,0,2\}$. We eventually present a
conjecture for the other eigenvalues and the degenerate Darboux point case
$V'(c)=0$.
