  Novel cluster spin model with interactions and disorder is introduced and
studied. In specific type of interactions, we find an extensive number of local
integrals of motion (LIOMs), which are a modified version of the stabilizers in
quantum information, i.e., mutually commuting operators specifying all quantum
states in the system. These LIOMs can be defined for any strength of the
interactions and disorder, and are of compact-support instead of
exponentially-decaying tail. Hence, even under the presence of interactions,
integrability is held, and all energy eigenstates are labeled by these LIOMs
and can be explicitly obtained. Integrable dynamics is, then, expected to
occur. The compact-support nature of the LIOMs crucially prevents the
thermalization and entanglement spreading. We numerically investigate dynamics
of the system governed by the existence of the compact-support LIOMs, and
clarify the effects of additional interactions, which break the compact-support
nature of the LIOMs. There, we find that the ordinary MBL behaviors emerge,
such as the logarithmic growth of the entanglement entropy in the time
evolution. Besides the ergodicity breaking dynamic, we find that
symmetry-protected-topological order preserves for specific states even in the
presence of the interactions.
