Stuart Vortices Extended to a Sphere Admits Infinite‐Dimensional Generalized Symmetries

To generalize Stuart vortices to the surface of a sphere, Crowdy recently obtained a nonlinear partial differential equation (involving a parameter γ) that has no known solution except for γ= 0 for which he gives a solution [ 1 ]. γ= 0 is therefore, assumed to be a kind of “solvability” condition for the equation. In this paper, to examine the integrability of this equation, we apply a generalized form of the Wahlquist–Estabrook prolongation procedure given in [ 2 ] to the equation for all nonzero values of γ . We see that the generalized symmetries of the Stuart vortices when extended to the surface of a sphere are infinite dimensional for all nonzero values of γ and are isomorphic to which becomes the minimal prolongation algebra of the equation.