On the integrability of certain matrix evolution equations

The matrix equation U = 2 U-3 + AU + UA is integrable there U = U(t) is a n x n-matrix, with n an arbitrary positive integer, and (A) under bar is an arbitrary constant n x n-matrix). The matrix evolution equation U = U-2 + a is also integrable (a arbitrary scalar constant). The matrix evolution equ ation U = f(U), where f((U) under bar) is an arbitrary function of (U) unde r bar (and of no other matrix, so that the commutator [(U) under bar,f((U) under bar)] vanishes) possesses at least n(2) - n (scalar) constants of mot ion. Lax pairs are exhibited for all these second-order n x n-matrix evolut ion ODEs. (C) 2000 Published by Elsevier Science B.V.