  We give a complete symbolic dynamics description of the dynamics of Euler's
problem of two fixed centers. By analogy with the 3-body problem we use the
collinearities (or syzygies) of the three bodies as symbols. We show that
motion without collision on regular tori of the regularised integrable system
are given by so called Sturmian sequences. Sturmian sequences were introduced
by Morse and Hedlund in 1940. Our main theorem is that the periodic Sturmian
sequences are in one to one correspondence with the periodic orbits of the two
center problem. Similarly, finite Sturmian sequences correspond to
collision-collision orbits.
