Abstract
The matrix equation U ̈ =2 U 3+AU+UA is integrable (here U= U( t) is a n× n-matrix, with n an arbitrary positive integer, and A is an arbitrary constant n× n-matrix). The matrix evolution equation U ̈ =U 2+a is also integrable ( a arbitrary scalar constant). The matrix evolution equation U ̈ =f(U) , where f U is an arbitrary function of U (and of no other matrix, so that the commutator U,f U vanishes) possesses at least n 2− n (scalar) constants of motion. Lax pairs are exhibited for all these second-order n× n-matrix evolution ODEs.
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