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.