We study two (massless free field) models, a photon/photino model with a vector gauge field and a Majorana spinor field, and a Wess-Zumino model. They each exhibit Lorentz symmetry violation but retain, in an appropriate way, the supersymmetry correspondance between the particles of the two fields. In relation to the photon field the Lorentz symmetry violation is of a simple but non-trivial kind that implies birefringence. In relation to the spinor field the Lorentz violation is produced by a modification of the Majorana equation that is a simplified version of more general investigations of Lorentz symmetry violation of the Dirac equation. In the case of the Wess-Zumino model we retain the same violation of Lorentz symmetry for the Majorana field and adjust the propagation of the scalar particles so that they exhibit a corresponding birefringence. The advantages of the models are that they are straightforward to investigate completely and both retain the basic aspect of supersymmetry namely the one-to-one correspondance between bosons and fermions. As a result of this bottom-up approach it is then possible to construct conserved supersymmetry charges and investigate their algebraic properties. To some extent these are similar to those encountered in the case of Lorentz invariance. However there are differences and in particular non-local terms appear in the commutation relations of the supersymmetry charges and fields of the models. We examine carefully the rather intricate nature of the limit back to Lorentz invariance.

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