Singularity analysis of a generalised Lagrange system

International audienceWe study the integrability of a Hamiltonian system proposed recently by Maciejewski and Przybylska, and which constitutes an extension of one studied (and integrated) by Lagrange. While the previous authors used the differential Galois theory formalism in order to study the integrability of this extended Hamiltonian, we approach the problem from the point of view of singularity analysis. For all values of the parameter (integer in the study of Maciejewski and Przybylska and rational in ours) we are able to show that the system does not possess the Painlevé property, with the exception of the case of the harmonic oscillator. The singularity structure of the Lagrange case is analysed in detail and commented upon