  Lorentz symmetry violation (LSV) can be generated at the Planck scale, or at
some other fundamental length scale, and naturally preserve Lorentz symmetry as
a low-energy limit (deformed Lorentz symmetry, DLS). DLS can have important
implications for ultra-high energy cosmic-ray physics (see papers
physics/0003080 - hereafter referred to as I -, astro-ph/0011181 and
astro-ph/0011182, and references quoted in these papers). A crucial question is
how DLS can be extended to a deformed Poincar\'e symmetry (DPS), and what can
be the dynamical origin of this phenomenon. In a recent paper (hep-th/0208064,
hereafter referred to as II), we started a discussion of proposals to identify
DPS with a symmetry incorporating the Planck scale (like doubly special
relativity, DSR) and suggested new ways in similar directions. Implications for
models of quadratically deformed relativistic kinematics (QDRK) and linearly
deformed relativistic kinematics (LDRK) were also discussed. We pursue here our
study of these basic problems, focusing on the possibility to relate deformed
relativistic kinematics (DRK) to new space-time dimensions and compare our QDRK
model, in the form proposed since 1997, which the Kirzhnits-Chechin (KCh) and
Sato-Tati (ST) models. It is pointed out that, although the KCh model does not
seem to work such as it was formulated, our more recent proposals can be
related to suitable extensions of this model generalizing the Finsler algebras
(even to situations where a preferred physical inertial frame exists) and using
the Magueijo-Smolin transformation as a technical tool.
