Symmetry analysis of ideal fluid equations in terms of trajectories and Weber's potential

The 2D perfect fluid motions equations in Lagrangian coordinates are considered. If body forces are potential one, then there is the general integral called Weber's integral and the resulting system includes initial data which in fact make the problem of group-theoretical classification actual. It is established that the basic group becomes infinite-dimensional with respect to the space variable too. The exceptional values of arbitrary initial vorticity are obtained at which we can be observed further extension of the group. Group properties of Euler equations in arbitrary Lagrangian coordinates are also considered and some exact solutions are constructed.