The Mixmaster Cosmological Model as a Pseudo-Euclidean Generalized Toda Chain

The question of the integrability of the mixmaster model of the Universe, presented as a dynamical system with finite degrees of freedom, is investigated in present paper. As far as the model belongs to the class of pseudo-Euclidean generalized Toda chains [1], the method of getting Kovalevskaya exponents developed in [2] for chains of Euclidean type, is used. The generalized formula of Adler and van Moerbeke [3] for systems of an indefinite metric is obtained. There was shown that although by the formula we got integer values of Kovalevskaya exponents there were multivalued solutions, branched at particular points on a plane of complex time t. This class of solutions differs from ones considered in [2]. Apparently, the system does not possess additional algebraic [3],[4] and one-valued [5] first integrals because of complex and transcendental values of the exponents. In addition, in the section 4 a ten-dimensional mixmaster model [6] was studied. There were not integer exponents.