Large-amplitude axisymmetric waves on columnar vortices, thought to be related to flow structures observed in vortex breakdown, are found as static bifurcations of the Bragg—Hawthorne equation. Solutions of this equation satisfy the steady, axisymmetric, Euler equations. Non-trivial solution branches bifurcate as the swirl ratio (the ratio of azimuthal to axial velocity) changes, and are followed into strongly nonlinear regimes using a numerical continuation method. Four types of solutions are found: multiple columnar solutions, corresponding to Benjamin’s ‘conjugate flows’, with subcritical—supercritical pairing of wave characteristics; solitary waves, extending previously known weakly nonlinear solutions to amplitudes large enough to produce flow reversals similar to the breakdown transition periodic wavetrains; and solitary waves superimposed on the conjugate flow that emerge from the periodic wavetrain as the wavelength or amplitude becomes sufficiently large. Weakly nonlinear soliton solutions are found to be accurate even when the perturbations they cause are fairly strong.