  Recent theoretical work has derived the correct form of the Ginzburg-Landau
differential equations, for the superconducting order parameter and vector
potential, in the presence of a small defect. Here, these equations are applied
to the case of a single vortex line pinned on such a defect. We develop the
coupled set of partial differential equations, and show how to derive analytic
solutions for the order parameter and magnetic field perturbations in the
region of space near the defect. Certain properties of the unperturbed vortex
solution are needed to totally specify our result; these are evaluated
numerically, and compared with those deduced from Clem's approximate solution.
