  We identify and study classes of initial states in integrable quantum systems
that, after the relaxation dynamics following a sudden quench, lead to
near-thermal expectation values of few-body observables. In the systems
considered here, those states are found to be insulating ground states of
lattice hard-core boson Hamiltonians. We show that, as a suitable parameter in
the initial Hamiltonian is changed, those states become closer to Fock states
(products of single site states) as the outcome of the relaxation dynamics
becomes closer to the thermal prediction. At the same time, the energy density
approaches a Gaussian. Furthermore, the entropy associated with the generalized
canonical and generalized grand-canonical ensembles, introduced to describe
observables in integrable systems after relaxation, approaches that of the
conventional canonical and grand-canonical ensembles. We argue that those
classes of initial states are special because a control parameter allows one to
tune the distribution of conserved quantities to approach the one in thermal
equilibrium. This helps in understanding the approach of all the quantities
studied to their thermal expectation values. However, a finite-size scaling
analysis shows that this behavior should not be confused with thermalization as
understood for nonintegrable systems.
