  We study a new non-local hierarchy of equations of the isentropic gas
dynamics type where the pressure is a non-local function of the density. We
show that the hierarchy of equations is integrable. We construct the two
compatible Hamiltonian structures and show that the first structure has three
distinct Casimirs while the second has one. The existence of Casimirs allows us
to extend the flows to local ones. We construct an infinite series of commuting
local Hamiltonians as well as three infinite series (related to the three
Casimirs) of non-local charges. We discuss the zero curvature formulation of
the system where we obtain a simple expression for the non-local conserved
charges, which also clarifies the existence of the three series from a Lie
algebraic point of view. We point out that the non-local hierarchy of
Hunter-Zheng equations can be obtained from our non-local flows when the
dynamical variables are properly constrained.
