  The currents and field distributions of a vortex in a thin superconducting
strip of a width $W$ is considered. It is shown that unlike infinite films
where the vortex field crosses the film only in one direction (say, from the
half-space under the film to the half-space above it), in strips (as well as in
finite size film samples of any shape) the field lines go back to the lower
half-space not only out of the sample but also through certain strip parts. The
strip patches where the magnetic lines ``dive" back to the space under the
strip are situated mostly near the strip edges at the strip and out of it. The
magnetic flux through the strip associated with the vortex is shown to be less
than the flux quantum and depends on the vortex position. The suppression of
the vortex flux is strong in narrow strips with $W<<\Lambda=2\lambda^2/d$ where
$\lambda$ is the bulk London penetration depth and $d$ is the film thickness.
The vortex energy scales roughly with $W/\Lambda$ and depends on vortex
position.
