The complete integrability of a Lie-Poisson system proposed by Bloch and Iserles

We establish the Liouville integrability of the differential equation $\dot S(t)= [N,S^2(t)],$ recently considered by Bloch and Iserles. Here, $N$ is a real, fixed, skew-symmetric matrix and $S$ is real symmetric. The equation is realized as a Hamiltonian vector field on a coadjoint orbit of a loop group, and sufficiently many commuting integrals are presented, together with a solution formula for their related flows in terms of a Riemann-Hilbert factorization problem. We also answer a question raised by Bloch and Iserles, by realizing the same system on a coadjoint orbit of a finite dimensional Lie group.