  We uncover the exact athermal eigenstates in the Bose-Hubbard (BH) model with
a three-body constraint, motivated by the exact construction of quantum
many-body scar (QMBS) states in the $S=1$ $XY$ model. These states are
generated by applying an $\rm SU(2)$ ladder operator consisting of a linear
combination of two-particle annihilation operators to the fully occupied state.
By using the improved Holstein-Primakoff expansion, we clarify that the QMBS
states in the $S=1$ $XY$ model are equivalent to those in the constrained BH
model with additional correlated hopping terms. We also find that, in the
strong-coupling limit of the constrained BH model, the QMBS state exists as the
lowest-energy eigenstate of the effective model in the highest-energy sector.
This fact enables us to prepare the QMBS states in a certain adiabatic process
and opens up the possibility of observing them in ultracold-atom experiments.
