A class of unsteady vortex solutions of the Euler equations is investigated. The solutions satisfy the von Kármán–Bödewadt similarity scalings and correspond to free and forced oscillations of radially unbounded solid-body-type vortices with axially varying rotation rates. The vortices may be of unbounded vertical extent or confined by impermeable top and/or bottom plates. In the latter case the bounding plates may be stationary or oscillatory. A solution breakdown result tends to support the hypothesis that breakdown of viscous Bödewadt-type counter-rotating vortex flows is an essentially inviscid process.