  In several experiments the magnitude of the contribution of magnetic
impurities to the Kondo resistivity shows size dependence in mesoscopic
samples. It was suggested ten years ago that magnetic surface anisotropy can be
responsible for the size dependence in cases where there is strong spin-orbit
interaction in the metallic host. The anisotropy energy has the form $\Delta
E=K_d ({\bf n}{\bf S})^2$ where ${\bf n}$ is the vector perpendicular to the
plane surface, ${\bf S}$ is the spin of the magnetic impurity and $K_d>0$ is
inversely proportional to distance $d$ measured from the surface. It has been
realized that in the tedious calculation an unjustified approximation was
applied for the hybridizations of the host atom orbitals with the conduction
electrons which depend on the position of the host atoms. Namely, the momenta
of the electrons were replaced by the Fermi momentum $k_F$. That is
reinvestigated considering the $k$-dependence which leads to singular energy
integrals and in contrary to the previous result $K_d$ is oscillating like
$\sin (2 k_F d)$ and the distance dependence goes like $1/d^3$ in the
asymptotic region. As the anisotropy is oscillating, for integer spin the
ground state is either a singlet or a doublet depending on distance $d$, but in
the case of the doublet there is no direct electron induced transition between
those two states at zero temperature. Furthermore, for half-integer ($S > 1/2$)
spin it is always a doublet with direct transition only in half of the cases.
