Nonintegrability of a class of the Bianchi V I0 and V II0 models

The well-known Bianchi V I0 and Bianchi V II0 dynamical systems are three-dimensional differential systems which after a convenient reduction become ẋ=−x2+(z+1)y2,ẏ=−4(z+1)+xyz,ż=−yz(z+2). In the paper of Maciejewski and Szydiowski [A.J. Maciejewski, M. Szydiowski, Bianchi cosmologies as dynamical systems, Celestial Mech. Dynam. Astronom. 73 (1999) 17–24], the authors asked about the integrability or nonintegrability of this system. Here we show that this system has no first integrals which are polynomial, rational, Darboux functions or analytic functions. Consequently this system is not integrable inside these classes of functions.