Third order integrability conditions for homogeneous potentials of degree -1

We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r, theta) = 1/r * h(exp(i*theta)) with h in C[z], deg(h) <= 3. We then find all meromorphically integrable potentials of this form.