  The problem of integrability of the mixmaster model as a dynamical system
with finite degrees of freedom is investigated. The model belongs to the class
of pseudo-Euclidean generalized Toda chains. It is presented as a
quasi-homogeneous system after transformations of phase variables. An
application of the method of getting of Kovalevskaya exponents to the model
leads to the generalized Adler -- van Moerbeke formula on root vectors. A
generalized Cartan matrix is constructed with use of simple root vectors in
Minkowski space. The mixmaster model is associated to a Borcherds algebra. The
known hyperbolic Kac -- Moody algebra of Chitre billiard model is obtained by
using three spacelike (without isotropic) root vectors.
