2000 Mathematics Subject Classification: 12D10.In the paper we give different examples of overdetermined strata.

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form $H=1/2{\sum }_{i=1}^{n}p{²}_i+V\left(q\right)$, where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form $V=V_k+V_K$, where $V_k$ and $V_K$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable...