  The vortex core structure in a d-wave superconductor is analyzed on the basis
of the quasi-classical Eilenberger theory beyond the Ginzburg-Landau framework.
The current and magnetic field distributions around an isolated vortex break
circular symmetry seen in s-wave pairing and show four-fold symmetry,
reflecting the internal degrees of freedom in d-wave pairing, {\it i.e.} $\hat
k_x^2-\hat k_y^2$ in reciprocal space through the low lying quasi-particle
excitations. The peculiar orientation of the flux line lattice observed
recently in a cuprate is argued in light of the present theory.
