  We study the dynamics of a tracer particle, which performs a totally directed
random walk in an adsorbed monolayer composed of mobile hard-core particles
undergoing continuous exchanges with a vapour phase. In terms of a
mean-field-type approach, based on the decoupling of the
tracer-particle-particle correlation functions into the product of pairwise,
tracer-particle correlations, we determine the density profiles of the
monolayer particles, as seen from the stationary moving tracer, and calculate
its terminal velocity, V_{tr}. In the general case the latter is determined
implicitly, as the solution of a certain transcendental equation. In two
extreme limits of slow and fast monolayer particles diffusion, we obtain
explicit asymptotic forms of V_{tr}. We show next that the density profile in
the monolayer is strongly inhomogeneous: In front of the stationary moving
tracer the local density is higher than the average value, \rho_L, and
approaches \rho_L as an exponential function of the distance from the tracer.
Past the tracer the local density is lower than \rho_L and the approach to
\rho_L may proceed differently depending whether the particles number in the
monolayer is not or is explicitly conserved. In the former case the approach is
described by an exponential dependence with a different characteristic length,
compared to the behavior in front of the tracer; in the latter case, the
density tends to \rho_L algebraically. The characteristic lengths and the
amplitudes of the density relaxation functions are also determined explicitly
