The Lagrangian dynamics of particles that move about in a time-independent circular symmetric geopotential field characterized by a single maximum at a particular radius is analyzed by transforming the Rotating Shallow Water Equations (RSWE) to an integrable, two-degrees-of-freedom (2DOF), system. The two conserved quantities along each particle trajectory are the total (kinetic and potential) energy and the angular momentum. The azimuthal (tangential) angle and the angular momentum constitute one pair of the conjugate variables. The steady state of this pair occurs only when the vortex rotates as a solid body at a frequency of f/2 (f is the Coriolis frequency). An analysis of the steady states of the (u, r) (where u is the radial velocity and r is the radius) pair of conjugate variable shows that this subsystem has a unique and uniform steady state with u=0 at the radius where the geopotential attains its maximal value and that this steady state is independent of the value of the geopotential’s magnitude. However, the nature of this steady state varies with the geopotential amplitude: For small geopotential amplitude this steady state is elliptic, and no additional steady state exists. In contrast, for large geopotential amplitude this steady state becomes hyperbolic and two additional, elliptic, steady states develop at larger and smaller radii. The pitchfork bifurcation at the radius of maximum geopotential that occurs when the geopotential amplitude is decreased has drastic consequences for particle trajectories in different heights of a hurricane-like vortex where the amplitude of the geopotential varies substantially with height, while the radius of this maximum changes only slightly. The net rate of azimuthal propagation (defined as the net change in azimuthal angle after the back-and-forth oscillations that result from the radial oscillation about the elliptic (u, r) steady state are averaged out) can also be estimated analytically and these estimates are validated by numerical solutions of the four non-linear RSWE. These results have important implications for the Lagrangian trajectories that can be expected to occur in different pressure levels in a hurricane and for the occurrence of outflows from the hurricane.