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A rotational fluid model which can be used to describe broad vortical flows ranging from large scale to the atmospheric mesoscale and the oceanic submesoscale is studied by the symmetry group theory. After introducing one scalar-, two vector-, and two tensor potentials, we find that the Lie symmetries of the extended system include many arbitrary functions of z and {z,t}. The obtained Lie symmetries are used to find some types of exact solutions. One of exact solutions can be used to qualitatively describe the three-dimensional structure of hurricanes.