  We obtain classical string solutions on RxS^2 by applying the dressing method
on string solutions with elliptic Pohlmeyer counterparts. This is realized
through the use of the simplest possible dressing factor, which possesses just
a pair of poles lying on the unit circle. The latter is equivalent to the
action of a single Backlund transformation on the corresponding sine-Gordon
solutions. The obtained dressed elliptic strings present an interesting
bifurcation of their qualitative characteristics at a specific value of a
modulus of the seed solutions. Finally, an interesting generic feature of the
dressed strings, which originates from the form of the simplest dressing factor
and not from the specific seed solution, is the fact that they can be
considered as drawn by an epicycle of constant radius whose center is running
on the seed solution. The radius of the epicycle is directly related to the
location of the poles of the dressing factor.
