  The spin and layer (pseudospin) degrees of freedom are entangled coherently
in the canted antiferromagnetic phase of the bilayer quantum Hall system at the
filling factor $\nu =2$. There emerges a complex Goldstone mode describing such
a combined degree of freedom. In the zero tunneling-interaction limit
($\Delta_{\text{SAS}}\rightarrow 0$), its phase field provokes a supercurrent
carrying both spin and charge within each layer. The Hall resistance is
predicted to become anomalous precisely as in the $\nu =1$ bilayer system in
the counterflow and drag experiments. Furthermore, it is shown that the total
current flowing in the bilayer system is a supercurrent carrying solely spins
in the counterflow geometry. It is intriguing that all these phenomena occur
only in imbalanced bilayer systems.
