  The implications of two different time constraints on the
Mathisson-Papapetrou-Dixon (MPD) equations are discussed under three spin
supplementary conditions (SSC). For this reason the MPD equations are revisited
without specifying the affine parameter and several relations are reintroduced
in their general form. The latter allows to investigate the consequences of
combining the Mathisson-Pirani (MP) SSC, the Tulczyjew-Dixon (TD) SSC and the
Ohashi-Kyrian-Semer\'{a}k (OKS) SSC with two affine parameter types: the proper
time on one hand and the parameterizations introduced in [Gen. Rel. Grav. 8,
197 (1977)] on the other. For the MP SSC and the TD SSC it is shown that
quantities that are constant of motion for the one affine parameter are not for
the other, while for the OKS SSC it is shown that the two affine parameters are
the same. To clarify the relation between the two affine parameters in the case
of the TD SSC the MPD equations are evolved and discussed.
