  For random quantum spin models, the strong disorder perturbative expansion of
the Local Integrals of Motion (LIOMs) around the real-spin operators is
revisited. The emphasis is on the links with other properties of the
Many-Body-Localized phase, in particular the memory in the dynamics of the
local magnetizations and the statistics of matrix elements of local operators
in the eigenstate basis. Finally, this approach is applied to analyze the
Many-Body-Localization transition in a toy model studied previously from the
point of view of the entanglement entropy.
