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The matrix equation U=2U 3 +AU+UA is integrable (here U=U(t) is a nxn-matrix, with n an arbitrary positive integer, and A is an arbitrary constant nxn-matrix). The matrix evolution equation U=U 2 +a is also integrable (a arbitrary scalar constant). The matrix evolution equation U=f(U), where fU is an arbitrary function of U (and of no other matrix, so that the commutator U,fU vanishes)...
Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion ('acceleration equal force'), with linear and cubic forces, in S-dimensional space (S=arbitrary positive integer, with special attention to S=1,2,3). For S>1 the equations of motion are written in covariant form (S-vector equal S-vector), entailing their rotational...
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