  We discuss the gauge theory for quantum group $SU_q (2)\times U(1) $ on the
quantum Euclidean space. This theory contains three physical gauge fields and
one$\; U(1)-$gauge field with a zero field strength. We construct the
quantum-group self-duality equation (QGSDE) in terms of differential forms and
with the help of the field-strength decomposition. A deformed analog of the
BPST-instanton solution is obtained. We consider a harmonic (twistor)
interpretation of QGSDE in terms of $SU_q (2)/U(1) $ quantum harmonics. The
quantum harmonic gauge equations are formulated in the framework of a
left-covariant 3D differential calculus on the quantum group $SU_q (2)$.
