  We study many-body localization (MBL) and delocalization from the perspective
of integrals of motion (IOMs). MBL can be understood phenomenologically through
the existence of macroscopically many localized IOMs. However, IOMs exist for
all many-body systems, and non-localized IOMs determine properties on the
ergodic side of the MBL transition too. Here we explore their properties using
our method of displacement transformations. We show how different quantities
can be calculated using the IOMs as an expansion in the number of operators.
For all values of disorder the typical IOMs are localized, suggesting the
importance of rare fluctuations in understanding the delocalization transition.
