  We consider the harmonic-superspace formalism in the $N=4$ supersymmetry
using the $SU(4)/SU(2)\times SU(2)\times U(1)$ harmonics which was earlier
applied to the abelian gauge theory. The N=4 non-abelian constraints in a
standard superspace are reformulated as the harmonic-superspace equations for
two basic analytic superfields: the independent superfield strength W of a
dimension 1 and the dimensionless harmonic gauge 4-prepotential V having the
$U(1)$ charge 2. These constraint equations I manifestly depend on the
Grassmann coordinates $\theta$, although they are covariant under the unusual
N=4 supersymmetry transformations. We analyze an alternative harmonic formalism
of the supergauge theory for two unconstrained nonabelian analytic superfields
W and V. The gauge-invariant action A(W,V) in this formalism contains $\theta$
factors in each term, it is invariant under the $SU(4)$ automorphism group. In
this model, the interaction of two infinite-dimensional N=4 supermultiplets
with the physical and auxiliary fields arises at the level of component fields.
The action A(W,V) generate analytic equations of motion II alternative to the
harmonic-superspace superfield constraints I. Both sets of equations give us
the equivalent equations for the physical component fields of the $N=4$ gauge
supermultiplet, they connect auxiliary and physical fields of two superfields.
The nonlinear effective interaction of the abelian harmonic superfield W is
constructed.
