Nonintegrability of a class of the Bianchi VI0 and VII0 models

The well-known Bianchi VI0 and Bianchi VII0 dynamical systems are three-dimensional differential systems which after a convenient reduction become (x) over dot = -x(2) + (z + 1)y(2), (y) over dot = -4(z + 1) + xyz, (z) over dot = -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. (C) 2010 Elsevier B.V. All rights reserved.