  We calculate exact one-dimensional collisionless plasma equilibria for a
continuum of flux tube models, for which the total magnetic field is made up of
the `force-free' Gold-Hoyle magnetic flux tube embedded in a uniform and
anti-parallel background magnetic field. For a sufficiently weak background
magnetic field, the axial component of the total magnetic field reverses at
some finite radius. The presence of the background magnetic field means that
the total system is not exactly force-free, but by reducing its magnitude the
departure from force-free can be made as small as desired. The distribution
function for each species is a function of the three constants of motion;
namely the Hamiltonian and the canonical momenta in the axial and azimuthal
directions. Poisson's Equation and Amp\`{e}re's Law are solved exactly, and the
solution allows either electrically neutral or non-neutral configurations,
depending on the values of the bulk ion and electron flows. These equilibria
have possible applications in various solar, space and astrophysical contexts,
as well as in the laboratory.
