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 $\scM_k\subset\Q$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$, belong to $\scM_k$. 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 $\scI_{n,k}\subset\scM_k$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$ belong to $\scI_{n,k}$. We give an algorithm which allows to find sets $\scI_{n,k}$. 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 non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.