  I show that the classical Toda models built on superalgebras, and obtained
from a reduction with respect to an $Sl(2)$ algebra, are "linearly
supersymmetrizable" (by adding spin 1/2 fields) if and only if the $Sl(2)$ is
the bosonic part of an $OSp(1|2)$ algebra. In that case, the model is
equivalent to the one constructed from a reduction with respect to the
$OSp(1|2)$ algebra, up to spin 1/2 fields. The corresponding $W$ algebras are
related through a factorization of spin 1/2 fields (bosons and fermions). I
illustrate this factorization on an example: the superconformal algebra built
on $Sl(n+1|2)$.
