Integrable homogeneous potentials of degree $-1$ in the plane with small eigenvalues

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