We study a recently proposed generalization of the relativistic particle by Kostelecký, that includes explicit Lorentz violation. We present an alternative action for this system and we show that this action can be interpreted as a particle in curved space with a metric that depends on the Lagrange multipliers. Furthermore, the following results are established for this model: (i) there exists a limit where this system has more local symmetries that the usual relativistic particle; (ii) in this limit if we restore the Lorentz symmetry we obtain a direct relationship with the two time physics; (iii) also we show that if we intent to restore the Poincaré symmetry we obtain the action of the relativistic bosonic string.

13 pages