  By using the moment algebra of the Vlasov kinetic equation, we characterize
the integrable Bloch-Iserles system on symmetric matrices
(arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the
corresponding Lie-Poisson structure by presenting a momentum map, which both
untangles the bracket structure and produces particle-type solutions that are
inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov
moments associated to Bloch-Iserles dynamics correspond to particular subgroup
inclusions into a group central extension (first discovered in
arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most
general case of Bloch-Iserles dynamics, a generalization of the Jacobi group
also emerges naturally.
