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 $V''(\bf d)$ calculated at a non-zero $\bf d \in \mathbb{C}^n$ satisfying $V'(\bf d) = \bf d$, 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 $V''(\bf d)$ 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.