The existence of multiple steady states is investigated in a simple photochemical reaction mechanism with the following properties: (1) it is derived from tropospheric chemistry, (2) it is governed by a nonlinear dynamic law, and (3) it exhibits temporal oscillations in the concentrations of its component chemical species. It is shown that chemical steady states for this mechanism can be located analytically using methods often also applicable to similar mechanisms. It is found that multiple steady states do not exist in this particular mechanism. Such an analytical treatment displays the dynamic structure of the mechanism, avoids numerical searching for steady states, and allows here for the identification and location of a transition (bifurcation) from steady-state to monotonic-growth behavior of [CO] and [O3] as the influx of NO increases at fixed values of the influxes of O3 and CO. Thus an atmosphere is seen to lose its ability to photochemically cleanse itself of CO at sufficiently high influx of NO. It is suggested that multiple steady states may be introduced into this mechanism by addition of transport or a well-known, hydrocarbon-based, radical-multiplication reaction.