Are non-dissipative hydrodynamic equations necessarily Hamiltonian?*

Isotropic fluids in two spatial dimensions can break parity symmetry and sustain transverse stresses which do not lead to dissipation. Corresponding transport coefficients include odd viscosity, odd torque and odd pressure. We consider the most general isotropic Galilean invariant fluid dynamics with momentum and particle density conservation. We limit ourselves to the terms of the second order within some counting scheme. We find exact conditions on transport coefficients which correspond to dissipationless and separately to Hamiltonian fluid dynamics. We find that the dissipationless fluids are not necessarily Hamiltonian. We discuss the consequences of this observation on the structure of effective fluid dynamics defined by reductions from more microscopically refined theories. As a consequence, we identify terms in the stress tensor that can only arise in out-of-equilibrium active matter.