The stability of fluid through a channel subject to a system rotation of constant rate about the spanwise axis is considered. In contrast to previous studies, a strongly nonlinear bifurcation approach is used to solve for a family of two-dimensional, steady, streamwise-orientated vortex flows. A stability analysis of these flows is also performed. All of the two-dimensional flows considered lose stability to an Eckhaus (streamwise-independent) secondary disturbance in a steady bifurcation to another member of the solution set. This property, given also that lower-order primary and secondary disturbance modes can become unstable, leads to a rich structure of bifurcation relationships between the secondary flows. With increasing Reynolds number, the secondary flow arising from the linear critical point first loses stability to the Eckhaus instability, and then loses stability to a fundamental spanwise mode with small streamwise wavenumber. With a further increase in Reynolds number, the secondary flow then also becomes unstable to a disturbance of subharmonic spanwise and $O(1)$ streamwise wavenumber, and finally (on the upper solution branch) a disturbance of fundamental spanwise and $O(1)$ streamwise wavenumber. Other types of bifurcation for possible tertiary flows are also identified. By superimposing the secondary disturbance onto the secondary flow, visualizations of the possible structure of the bifurcating tertiary flows are obtained. The visualizations show low-speed streaks in the streamwise velocity component lying between a set of staggered vortices for superharmonic bifurcations, and between aligned vortices for subharmonic bifurcations. Excellent qualitative and quantitative agreement is found with previous experimental results and direct-numerical-simulation-based stability studies, and good overall agreement with previous DNS studies was also found.