  Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of
pseudofermions, was introduced in the context of a new Kemmer equation
describing charged vector mesons interacting with an external constant magnetic
field. Here we construct the complete explicit solution for its realization in
terms of two superpotentials, both equal or unequal. We prove that any
orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and
give conditions under which a pseudosupersymmetric one may be described by
orthosupersymmetries of order two. We propose two new matrix realizations of
PsSSQM in terms of the generators of a generalized deformed oscillator algebra
(GDOA) and relate them to the cases of equal or unequal superpotentials,
respectively. We demonstrate that these matrix realizations are fully reducible
and that their irreducible components provide two distinct sets of bosonized
operators realizing PsSSQM and corresponding to nonlinear spectra. We relate
such results to some previous ones obtained for a GDOA connected with a
$C_3$-extended oscillator algebra (where $C_3 = {\rm Z}_3$) in the case of
linear spectra.
