  The transverse velocities of steadily rotating, non--axisymmetric patterns in
flat galaxies may be determined by a purely kinematical method, using two
dimensional maps of a tracer surface brightness and radial current density. The
data--maps could be viewed as the zeroth and first velocity moments of the
line--of--sight velocity distribution, which is the natural output of
integral--field spectrographs. Our method is closely related to the
Tremaine--Weinberg method of estimating pattern speeds of steadily rotating
patterns, when the tracer surface brightness satisfies a source--free
continuity equation. We prove that, under identical assumptions about the
pattern, two dimensional maps may be used to recover not just one number (the
pattern speed), but the full vector field of tracer flow in the disc plane. We
illustrate the recovery process by applying it to simulated data, and test its
robustness by including the effects of noise.
