"Determinant" as a Hamiltonian

Hi
consider the Hamiltonian H=xw-yz defined on R^4,the hamiltonian vector field is x'=-y y'=x z'=-w w'=-z
obviously H'=0 and G'=0 where G=x^2+y^2+z^2+w^2
and gradients of G and H are independent vectors:in fact
if we identify R^4 with M_2(R),then "Determinant" is a Completly Integrable hamiltonian
My Question :Is "Determinant" Completly integrable in arbitrary dimension(consider M_n(R) where n is even)
so we have a vector field on R^(n^2),and we search for n^2/2 number of independent maps G with G'=0
what information and resuluts do exist about this special Hamiltonian?
Thank you
Ali Taghavi

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