Motivated by observations of Williamson & Koshko of the wake of an osc
illating cylinder with three vortices per cycle, and by the analyses o
f Rott and Aref of the motion of three vortices with vanishing net cir
culation on the unbounded plane, the integrable problem of three inter
acting, periodic vortex rows is solved. The problem is 'mapped' onto a
problem of advection of a passive particle by a certain set of fixed
point vortices. The results of this mapped problem are then re-interpr
eted in terms of the motion of the vortices in the original problem. A
rather complicated structure of the solution space emerges with a sur
prisingly large number of regimes of motion, some of them somewhat cou
nter-intuitive. Representative cases are analysed in detail, and a gen
eral procedure is indicated for all cases. We also trace the bifurcati
ons of the solutions with changing linear momentum of the system. For
rational ratios of the vortex circulations all motions are periodic. F
or irrational ratios this is no longer true. The point vortex results
are compared to the aforementioned wake experiments and appear to shed
light on the experimental observations. Many additional possibilities
for the wake dynamics are suggested by the analysis.