A nonlinear analysis of parameter regions in the “two-temperature” reactor stability problem is accomplished using methods developed in the USSR for treating ordinary differential equations. It is shown that in a model where both temperature-dependent quantities obey Newton's law of cooling, stable limit cycles exist and centers do not exist. If one of the quantities obeys an adiabatic cooling law, centers exist and stable limit cycles do not exist. Solutions with finite escape time are found to exist for certain sets of parameters and initial conditions. Finally, when at least one linear characteristic root vanishes, it is shown that a first integral exists and that it is possible to discuss reactor behavior in terms of this integral.