  For random interacting Majorana models where the only symmetries are the
Parity $P$ and the Time-Reversal-Symmetry $T$, various approaches are compared
to construct exact even and odd normalized zero modes $\Gamma$ in finite size,
i.e. hermitian operators that commute with the Hamiltonian, that square to the
Identity, and that commute (even) or anticommute (odd) with the Parity $P$.
Even Normalized Zero-Modes $\Gamma^{even}$ are well known under the name of
'pseudo-spins' $\tau^z_n$ in the field of Many-Body-Localization or more
precisely 'Local Integrals of Motion' (LIOMs) in the Many-Body-Localized-Phase
where the pseudo-spins happens to be spatially localized. Odd Normalized
Zero-Modes $\Gamma^{odd}$ are popular under the name of 'Majorana Zero Modes'
or 'Strong Zero Modes'. Explicit examples for small systems are described in
detail. Applications to real-space renormalization procedures based on blocks
containing an odd number of Majorana fermions are also discussed.
