  We derive the equations of motion for a system undergoing boost-invariant
longitudinal and azimuthally-symmetric transverse "Gubser flow" using
leading-order anisotropic hydrodynamics. This is accomplished by assuming that
the one-particle distribution function is ellipsoidally-symmetric in the
momenta conjugate to the de Sitter coordinates used to parameterize the Gubser
flow. We then demonstrate that the SO(3)_q symmetry in de Sitter space further
constrains the anisotropy tensor to be of spheroidal form. The resulting system
of two coupled ordinary differential equations for the de Sitter-space momentum
scale and anisotropy parameter are solved numerically and compared to a
recently obtained exact solution of the relaxation-time-approximation Boltzmann
equation subject to the same flow. We show that anisotropic hydrodynamics
describes the spatio-temporal evolution of the system better than all currently
known dissipative hydrodynamics approaches. In addition, we prove that
anisotropic hydrodynamics gives the exact solution of the relaxation-time
approximation Boltzmann equation in the ideal, eta/s -> 0, and free-streaming,
eta/s -> infinity, limits.
