Partial and super-integrability of Hamiltonian systems. A differential Galois approach.

We consider natural Hamiltonian systems with homogeneous potential and
$n$ degrees of freedom. We give computable necessary conditions for:

1. the existence of $k$ commuting and independent first integrals
where $1leq k leq n$.
2. the existence of $n+k$ independent first
integrals such that $n$ of them commute and $1leq k leq n-1$.

Our results generalize the well known Morales-Ramis theorem which gives
necessary conditions for the Liouville integrability of the considered
class of Hamiltonian systems.