Justification of Painlevé analysis for Hamiltonian systems by differential Galois theory

The discovery of the Kowalevski top (1889) as a new integrable system posed the question whether there exists a rigorous relation between integrability of a Hamiltonian system and the analytic property of the solution in the complex time plane. Many examples suggest a hidden relation between the nature of singularities of solution, and the integrability of the system. Without enough justification of the method itself, this so-called Painlevé analysis (to determine the values of parameters such that the singularities are only poles) made it possible to discover some new integrable systems. In this paper, a recent justification of this analysis will be reviewed which is based on the differential Galois theory. A rigorous statement is that possessing the weak Painlevé property is a necessary condition for integrability of Hamiltonian systems with a homogeneous potential.