In this work the nonlinear behavior of a coupled reactor-separator network is analyzed using singularity and bifurcation theory. The reactor is modeled as a CSTR, which sustains an exothermic first-order reaction. The separator is modeled as a flash process. The effluent from the reactor is fed to the separator. The reactant-rich stream (assumed to be the bottoms) from the separator is recycled back to the reactor. Focus is on pure mass recycle, as the two units are decoupled energetically via heat exchangers. Two different modes of operation are considered. These are compared with each other and with the well-known stand alone reactor with a first-order exothermic reaction. For the first mode of operation, the feed rate is fixed and for the second the recycle rate is fixed. In practice, these different modes of operation can be achieved by a suitable control strategy. It is shown that the behavior crucially depends on the mode of operation. Fixing the feed rate can lead to severe operational problems including monotonic and oscillatory unstable steady states over a wide range of operating conditions. Further, parameter regions are identified, where no steady state exists at all. On the other hand when the recycle rate is fixed the coupled system admits at least one stable steady-state solution for a fixed set of operating conditions.