Integrable hydrodynamic chains associated with Dorfman Poisson brackets

This paper is devoted to a description of integrable Hamiltonian hydrodynamic chains associated with Dorfman Poisson brackets. Three main classes of these hydrodynamic chains are selected. Generating functions of conservation laws and commuting flows are found. Hierarchies of these Hamiltonian hydrodynamic chains are extended on negative moments and negative time variables. Corresponding three dimensional quasilinear equations of the second order are presented.