The behaviour of a bounded radiating body in general relativity is determined by a nonlinear boundary condition at the stellar surface. We systematically study the differential equations that arise using the Lie symmetry infinitesimal generators. We show that several nonlinear equations, including Bernoulli equations and Abel equations of the second kind, in addition to Riccati equations, are generated by assuming functional relationships on the gravitational potentials. We demonstrate that these equations may be solved exactly. The models found admit a linear equation of state for the radial pressure and the energy density. The energy conditions are satisfied and the matter variables are well behaved.