  We construct a manifestly $N=(4,0)$ world-sheet supersymmetric twistor-like
formulation of the $D=6$ Green-Schwarz superstring, using the principle of
double (target-space and world-sheet) Grassmann analyticity. The superstring
action contains two Lagrange multiplier terms and a Wess-Zumino term. They are
written down in the analytic subspace of the world-sheet harmonic $N=(4,0)$
superspace, the target manifold being too an analytic subspace of the harmonic
$D=6\;\; N=1$ superspace. The kappa symmetry of the $D=6$ superstring is
identified with a Kac-Moody extension of the world-sheet $N=(4,0)$
superconformal symmetry. It can be enlarged to include the whole world-sheet
reparametrization group if one introduces the appropriate gauge Beltrami
superfield into the action. To illustrate the basic features of the new $D=6$
superstring construction, we first give some details about the simpler (already
known) twistor-like formulations of $D=3, N=(1,0)$ and $D=4, N=(2,0)$
superstrings.
