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We consider natural complex Hamiltonian systems with $n$ degrees of freedomgiven by a Hamiltonian function which is a sum of the standard kinetic energyand a homogeneous polynomial potential $V$ of degree $k>2$. The well knownMorales-Ramis theorem gives the strongest known necessary conditions for theLiouville integrability of such systems. It states that for each $k$ thereexists an explicitly known infinite set $\scM_k\subset\Q$ such that if thesystem 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. Undercertain genericity assumption concerning $V$ we prove the following fact. Foreach $k$ and $n$ there exists a finite set $\scI_{n,k}\subset\scM_k$ such thatif 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 findsets $\scI_{n,k}$. We applied this results for the case $n=k=3$ and we foundall integrable potentials satisfying the genericity assumption. Among themseveral are new and they are integrable in a highly non-trivial way. We foundthree potentials for which the additional first integrals are of degree 4 and 6with respect to the momenta.

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