  Hilbert space fragmentation is an ergodicity breaking phenomenon, in which
Hamiltonian shatters into exponentially many dynamically disconnected sectors.
In many fragmented systems, these sectors can be labelled by statistically
localized integrals of motion, which are nonlocal operators. We study the
paradigmatic nearest-neighbor pair hopping (PH) model exhibiting the so-called
strong fragmentation. We show that this model hosts local integrals of motion
(LIOMs), which correspond to frozen density modes with long wavelengths. The
latter modes become subdiffusive when longer-range pair hoppings are allowed.
Finally, we make a connection with a tilted (Stark) chain. Contrary to the
dipole-conserving effective models, the tilted chain is shown to support either
Hamiltonian or dipole moment as an LIOM. Numerical results are obtained from a
numerical algorithm, in which finding LIOMs is reduced to the data compression
problem.
