  We work on some general extensions of the formalism for theories which
preserve the relativity of inertial frames with a nonlinear action of the
Lorentz transformations on momentum space. Relativistic particle models
invariant under the corresponding deformed symmetries are presented with
particular emphasis on deformed dilatation transformations. The algebraic
transformations relating the deformed symmetries with the usual (undeformed)
ones are provided in order to preserve the Lorentz algebra. Two distinct cases
are considered: a deformed dilatation transformation with a spacelike preferred
direction and a very special relativity embedding with a lightlike preferred
direction. In both analysis we consider the possibility of introducing quantum
deformations of the corresponding symmetries such that the spacetime
coordinates can be reconstructed and the particular form of the real
space-momentum commutator remains covariant. Eventually feasible experiments,
for which Lorentz violating effects here pointed out may be detectable, are
suggested.
