  We obtained new periodic solutions in the problems of three and four point
vortices moving on a plane. In the case of three vortices, the system is
reduced to a Hamiltonian system with one degree of freedom, and it is
integrable. In the case of four vortices, the order is reduced to two degrees
of freedom, and the system is not integrable. We present relative and absolute
choreographies of three and four vortices of the same intensity which are
periodic motions of vortices in some rotating and fixed frame of reference,
where all the vortices move along the same closed curve. Similar choreographies
have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the
n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the
choreographies that appear in vortex dynamics have a number of distinct
features.
