The nonlinear dynamo wave equation, which has been formulated to explore oscillating dynamos, is found also to have steady magnetic field condfigurations as its stable solutions. The solutions of the nonlinear wave equation, integrated numerically as the initial-boundary-value problem in the rotating spherical geometry, eventually bifurcate into a stationary oscillating state and a stationary steady state, depending on the initial condition adopted in the integration. Both states are stable with respect to small perturbations. In the steady-state solutions, the magnetic configuration is that of a helical tube so that the dynamo process, being controlled by the nonlinear process, adjusts itself to be exactly balanced with the diffusion process. The relative sensitivity of the bifurcation of the system depends on the structure of the dynamo system and the strength of the nonlinear process. We suggest that the magnetic fields of the Earth and planets, and the fields of non--solar-type magnetic stars, especially stars classified as oblique rotators, can be understood as special stationary solutions of the nonlinear dynamo wave equation, which can also have oscilating solutions. Thus the field reversal of so-called steady dynamos can be understood naturally as the transition governed by the wave nature of the equation between themore » two stationary states when some change occurs temporarily in the dynamics of the dynamos.« less