  We discuss the left-covariant 3-dimensional differential calculus on the
quantum sphere $SU_q (2)/U(1) $. The $SU_q (2)$-spinor harmonics are treated as
coordinates of the quantum sphere. We consider the gauge theory for the quantum
group $SU_q (2)\times U(1) $ on the deformed Euclidean space $E_q (4)$. A
$q$-generalization of the harmonic-gauge-field formalism is suggested . This
formalism is applied for the harmonic (twistor) interpretation of the
quantum-group self-duality equation (QGSDE). We consider the zero-curvature
representation and the general construction of QGSDE-solutions in terms of the
analytic prepotential.
