  The gravitational field exterior respectively interior to an axially
symmetric, metrically stationary, isolated spinning source made of perfect
fluid is examined within the quasi-metric framework. (A metrically stationary
system is defined as a system which is stationary except for the direct effects
of the global cosmic expansion on the space-time geometry.) Field equations are
set up and an attempt is made to find an approximate series solution for the
exterior part. However, the result is that no stationary solution corresponding
to a spinning source can exist when considering terms beyond a certain order in
small quantities. That is, except for metrically static systems, axially
symmetric systems must necessarily be non-stationary in quasi-metric gravity.
However, sufficiently weak, axially symmetric gravitational fields associated
with slowly rotating sources, may still be considered as stationary as an
excellent approximation.
  Thus a truncated, approximately stationary solution is found for the exterior
field. To lowest order in small quantities, the gravito-magnetic part of the
found metric family corresponds with the Kerr metric in the metric
approximation. On the other hand, the gravito-electric part of the found metric
family also includes a tidal term characterized by the free quadrupole-moment
parameter $J_2$ describing the effect of source deformation due to the
rotation. This term has no counterpart in the Kerr metric. Finally, the
geodetic effect for a gyroscope in orbit is calculated. There is a correction
term, unfortunately barely too small to be detectable by Gravity Probe B, to
the standard expression.
