A generalization of the Lax equation defined by an arbitrary anti-automorphism

We consider a class of evolution equations on the Lie group GL(n, R) or any of its closed subgroups, built by means of an arbitrary anti-automorphism of the associative algebra of all real n-dimensional matrices M-n x n. The set of first integrals and a method of construction for a Hamiltonian subcl ass is shown. This subclass has a connection with the factorization problem . A certain application of a matrix evolution equation built by means of tr ansposition, related to the existence of (2, 0)- and (0, 2)-type tensor inv ariants in the theory of dynamical systems, is found.