  A new proof is given of the fact that the particle trajectories of the ideal
incompressible fluid are analytic curves, though the solutions of the Euler
equations may have a finite regularity. This is a consequence of a general fact
that the geodesic exponential map on the group of volume preserving
diffeomorphisms belonging to the Sobolev space is real-analytic. The proof is
based on the general properties of holomorphic maps in complex Banach spaces.
