This paper provides an analysis of the time evolution of a many-particle system starting out of equilibrium with its control parameter fixed at a value corresponding to a many-body energy-level crossing (degeneracy). We prove theorems concerning ergodicity, equilibration, and thermalization. For certain conditions, the occupancy of symmetrically equivalent basis states has different time-averaged probabilities. This nonergodicity remains in equilibrium. If the symmetrically equivalent states have opposite parity in relation to some physical property, then a left and right particle number imbalance averaged in time is nonzero. This imbalance does not occur for all initial basis states. In addition, the Hilbert space of the system is not fragmented; however, there is a subspace spanned by favored basis states, where the system is most likely to be found. Therefore, our results reveal what appears to be a unique mechanism for a weak eigenstates-thermalization-hypothesis breakdown, where the degenerate eigenstates can work as nonthermal eigenstates. To illustrate these findings, we consider the Hubbard Hamiltonian. In this case, ergodicity breaking produces a left and right magnetization imbalance, where the time-averaged probability of finding a spin-σ electron on one side of the crystal lattice is greater than on the other side. This imbalance is not associated with electrical charge; thus the conductance is preserved. The potential use in technology is discussed.