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 ${\mathcal M}_k\subset{\mathbb Q}$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''({\boldsymbol d})$ calculated at a non-zero ${\boldsymbol d}\in{\mathbb C}^n$ satisfying $V'({\boldsymbol d})={\boldsymbol d}$, belong to ${\mathcal M}_k$.
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 ${\mathcal I}_{n,k}\subset{\mathcal M}_k$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''({\boldsymbol d})$ belong to ${\mathcal I}_{n,k}$. We give an algorithm which allows to find sets ${\mathcal I}_{n,k}$.
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.