Hydromagnetic flow in the Taylor-Couette system with an electrically conducting fluid, an axial magnetic field, and counterrotating cylinders, is studied. Parameter combinations called bicritical points such that the basic Couette flow loses stability simultaneously due to axisymmetric (m = 0) and non-axisymmetric (m = 1) linear perturbations, are located. Nonlinear analysis shows the existence of Taylor vortices, spiral vortices, ribbons, wavy vortices and twisted vortices, for parameters near such bicritical points.; For zero magnetic field, results of Langford et al.{dollar}sp1{dollar} and of Golubitsky and Langford{dollar}sp2{dollar} are extended to smaller radius ratios, a new, third bicritical point is found and the structure of the families of solutions near this point is investigated. All three bicritical points and the corresponding families of solutions persist for nonzero values of the magnetic field, and increasing the magnetic field has an effect similar to decreasing the radius ratio. The results for nonzero magnetic field assume insulating boundary conditions.; Bistability is found in the neighborhood of all bicritical points located: for certain parameter ranges, there exist two families of solutions, both potentially stable.
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