Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

We consider natural complex Hamiltonian systems with n degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential V of degree k > 2. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each k there exists an explicitly known infinite set Mk ⊂ Q such that if the system is integrable, then all eigenvalues of the Hessian matrix V′′(d) calculated at a non-zero d ∈ Cn satisfying V′(d) = d, belong to Mk. The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning V we prove the following fact. For each k and n there exists a finite set In,k ⊂ Mk such that if the system is integrable, then all eigenvalues of the Hessian matrix V′′(d) belong to In,k. We give an algorithm which allows to find sets In,k. We applied this results for the case n = k = 3 and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly nontrivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta. MSC2000 numbers: 37J30, 70H07, 37J35, 34M35.