  We consider solutions with conformal symmetry of the static, spherically
symmetric gravitational field equations in the vacuum in the brane world
scenario. By assuming that the vector field generating the symmetry is
non-static, the general solution of the field equations on the brane can be
obtained in an exact parametric form, with the conformal factor taken as
parameter. As a physical application of the obtained solutions we consider the
behavior of the angular velocity of a test particle moving in a stable circular
orbit. In this case the tangential velocity can be expressed as a function of
the conformal factor and some integration constants only. For a specific range
of the integration constants, the tangential velocity of the test particle
tends, in the limit of large radial distances, to a constant value. This
behavior is specific to the galactic rotation curves, and is explained usually
by invoking the hypothesis of the dark matter. The limiting value of the
angular velocity of the test particle can be obtained as a function of the
baryonic mass and radius of the galaxy. The behavior of the dark radiation and
dark pressure terms is also considered in detail, and it is shown that they can
be expressed in terms of the rotational velocity of a test particle. Hence all
the predictions of the present model can be tested observationally. Therefore
the existence of the non-local effects, generated by the free gravitational
field of the bulk in a conformally symmetric brane, may provide an explanation
for the dynamics of the neutral hydrogen clouds at large distances from the
galactic center.
