  In the present work, the exact solution of Einstein's field equations which
has been given by Curzon in 1924 representing the field of a static binary
system is reviewed. An adapted version of this solution is obtained to describe
a dynamical binaries in a rotating coordinate system. It is shown that this
version of the solution is time-dependent. It reduces to the later one in the
static case if the rotation goes to zero. The original Curzon solution shows
that there are two singularities at the two masses, while in the modified
version the singularities become on the world-line of the two masses. The
solution shows no additional coordinate singularities. The killing vector field
of the axial symmetry is obtained in the modified version. In addition, the
rotation admits a further rotational symmetry, so a rotation killing vector
field is also obtained and discussed. The equations of motion for a test
particle in the field of a binary system are formulated and solved. Such
equations have been used to study the gravitational time delay of arrival
(Shapiro delay) of signals from binary pulsar systems resulted from our
suggested modifications containing additional terms. These terms are
interpreted as higher order corrections to the masses. In particular we
investigate the gravito-magnetic effect due to orbital angular motion of the
double pulsars. We give numerical estimates of this type of the time delay in
the case of the double-pulsar system PSR J0737-3039 A/B. We draw a model curve
for the gravito-magnetic time delay during one orbital revolution. We suggest
that this type of delay will have a larger contribution during the last phase
of the system evolution.
