Summary: We comment on an analysis by Contopoulos et al. which demonstrates that the governing six-dimensional Einstein equations for the mixmaster spacetime metric pass the ARS or reduced Painlevé test. We note that this is the case irrespective of the value, \(I\), of the generating Hamiltonian which is a constant of motion. For \(I < 0\) we find numerous closed orbits with two unstable eigenvalues strongly indicating that two additional first integrals apart from the Hamiltonian cannot exist and thus that the system, at least for this case, is very probably not integrable. In addition, we present numerical evidence that the average Lyapunov exponent nevertheless vanishes.
The model is thus a very interesting example of a Hamiltonian dynamical system, which is probably nonintegrable yet passes the reduced Painlevé test.