We consider a highly exothermic reaction being carried out in a long tubular fixed-bed reactor, whose walls are maintained at a fixed temperature T//c by external cooling. To maintain control of these reactions and avoid reactor runaways, such reactors must be designed so that their temperature rises are small fractions of the adiabatic temperature rise which would occur if the reactor was insulated. A set of reactor equations are developed under certain conditions, and we construct the bifurcation diagram for the radially-symmetric steady solutions, using T//c as the bifurcation parameter. We find that the only bifurcation points are turning points. We then analyze the stability and domains-of-attraction of these symmetric steady solutions. We discover that only the bottom solution branch represents practical operating states for the reactor: all other steady solutions are either unstable or occur at unreasonably high temperatures.