I will present an overview of my works connected with the integrability of natural Hamiltonian system with homogeneous potentials. An application of differential Galois methods for such system is effective as we known for them particular solutions. These solutions are defined by an algebraic set $\mathcal{D}$ in a complex projective space. It appears that residue of certain differential forms taken over points of $\mathcal{D}$ restricts the necessary conditions for the integrability deduced from the differential Galois theory.