  Even though foundations of the eigenstate thermalization hypothesis (ETH) are
based on random matrix theory, physical Hamiltonians and observables
substantially differ from random operators. One of the major challenges is to
embed local integrals of motion (LIOMs) within the ETH. Here we focus on their
impact on fluctuations and structure of the diagonal matrix elements of local
observables. We first show that nonvanishing fluctuations entail the presence
of LIOMs. Then we introduce a generic protocol to construct observables,
subtracted by their projections on LIOMs as well as products of LIOMs. The
protocol systematically reduces fluctuations and/or the structure of the
diagonal matrix elements. We verify our arguments by numerical results for
integrable and nonintegrable models.
