A full symmetry classification is given for models of energy transport in radiant plasma when the mass density is spatially variable and the diffusivity is nonlinear. A systematic search for conservation laws also leads to some potential symmetries, and to an integrable nonlinear model. Classical point symmetries, potential symmetries and nonclassical symmetries are used to effect variable reductions and exact solutions. The simplest time-dependent solution is shown to be stable, and relevant to a closed system. ; Comment: 13 pages. arXiv admin note: text overlap with arXiv:math-ph/0605081, arXiv:1101.4755 by other authors