The Michelson system is neither global analytic, nor Darboux integrable

We consider the differential system ẋ=y, ẏ=z, ż=c2−y−x2/2 in R3R3, where cc is a real parameter. This differential system is known as the Michelson system and its dynamics has been studied during these last twenty five years but nothing was known up to now on its integrability. We show that for any value of cc the Michelson system is neither global analytic, nor Darboux integrable.