  We study ergodicity breaking in the clean Bose-Hubbard chain for small
hopping strength. We see the existence of a non-ergodic regime by means of
indicators as the half-chain entanglement entropy of the eigenstates, the
average level spacing ratio, {the properties of the eigenstate-expectation
distribution of the correlation and the scaling of the Inverse Participation
Ratio averages.} We find that this ergodicity breaking {is different from
many-body localization} because the average half-chain entanglement entropy of
the eigenstates obeys volume law. This ergodicity breaking appears unrelated to
the spectrum being organized in quasidegenerate multiplets at small hopping and
finite system sizes, so in principle it can survive also for larger system
sizes. We find that some imbalance oscillations in time which could mark the
existence of a glassy behaviour in space are well described by the dynamics of
a single symmetry-breaking doublet and {quantitatively} captured by a
perturbative effective XXZ model. We show that the amplitude of these
oscillations vanishes in the large-size limit. {Our findings are numerically
obtained for systems with $L < 12$. Extrapolations of our scalings to larger
system sizes should be taken with care, as discussed in the paper.
