Spherical flame balls are studied using a model for the chemical kinetics which involves a non-exothermic autocatalytic reaction, describing the chain-branching generation of a chemical radical and an exothermic completion reaction, the rate of which does not depend on temperature. When the chain-branching reaction has a large activation temperature, an asymptotic structure emerges in which the branching reaction generates radicals and consumes fuel at a thin flame interface, although heat is produced and radicals are consumed on a more distributed scale. Another model, based more simply, but less realistically, on the generation of radicals by decomposition of the fuel, provides exactly the same leading order matching conditions. These can be expressed in terms of jump conditions across a reaction sheet that are linear in the dependent variables and their normal gradients. Using these jump conditions, a reactive-diffusive model with linear heat loss then leads to analytical solutions that are multivalued for small enough levels of heat loss, having either a larger or a smaller radius of the interface where fuel is consumed. The same properties are found, numerically, to persist as the activation temperature of the branching reaction is reduced to values that seem to be typical for hydrocarbon chemistry. Part of the solution branch with larger radius is shown to become stable for low enough values of the Lewis number of the fuel.