We characterize a class of integrable Hamiltonian hydrodynamic chains, based on the
necessary condition for the integrability provided by the vanishing of the Haantjes tensor.
We prove that the vanishing of the first few components of the Haantjes tensor is already
sufficiently restrictive, and allows a complete description of the corresponding Hamiltonian
densities. In each of the cases we were able to explicitly construct a generating function for
conservation laws, thus establishing the integrability.