The geodesic Vlasov equation and its integrable moment closures

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles. Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context