  We exploit a generic instability in the integration of steady, sonic,
near-isothermal flows to find the complete transition diagram for recombination
fronts (for a model system of equations). The instability requires the
integration of the flow equations for speeds between the isothermal and
adiabatic sound speeds to be performed with particular care. As a result of
this, the previous work of Newman & Axford on the structure of recombination
fronts neglected an important class of solution, that of transonic fronts; our
method is readily extensible to a more complete treatment of the ionization
structure. Future papers will apply these results in models of the structure of
ultracompact HII regions.
