The weak, dominant and strong energy conditions are investigated for various kinds of imperfect fluids. In this context, attention has been given to the model of a collapsing or expanding sphere of shear-free fluid which conducts heat and radiates energy to infinity.

We consider lattice equations on Z 2 which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three-and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogues of the Painlevé equations are considered.

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