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

We establish the Liouville integrability of the differential equation Ṡ(t) = [N,S²(t)], recently considered by Bloch and Iserles. Here, N is a real, fixed, skew-symmetric matrix and S is a 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.