  The Euler equation (EE) is one of the basic equations in many physical fields
such as the fluids, plasmas, condense matters, astrophysics, oceanic and
atmospheric dynamics. A new symmetry group theorem of the two dimensional EE is
obtained via a simple direct method and the theorem is used to find \em exact
analytical \rm vortex and circumfluence solutions. Some types of Darboux
transformations (DTs) for the both two and three dimensional EEs are obtained
for \em arbitrary spectral parameters \rm which indicates that the EEs are
integrable and the Navier-Stockes (NS) equations with large Renoyed number are
nearly integrable, i.e, they are singular perturbations of the integrable EEs.
The possibility of the vortex and circumfluence solutions to approximately
explain the tropical cyclones (TCs), especially, the Hurricane Andrew 1992, is
discussed.
