We develop a novel bi-harmonic $$ \mathcal{N} $$ N = 4 superspace formulation of the $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory (SYM) in four dimensions. In this approach, the $$ \mathcal{N} $$ N = 4 SYM superfield constraints are solved in terms of on-shell $$ \mathcal{N} $$ N = 2 harmonic superfields. Such an approach provides a convenient tool of constructing the manifestly $$ \mathcal{N} $$ N = 4 supersymmetric invariants and further rewriting them in $$ \mathcal{N} $$ N = 2 harmonic superspace. In particular, we present $$ \mathcal{N} $$ N = 4 superfield form of the leading term in the $$ \mathcal{N} $$ N = 4 SYM effective action which was known previously in $$ \mathcal{N} $$ N = 2 superspace formulation.

We present the explicit superfield realizations of the hidden SU (4) and O(5) Rsymmetries in 4D, N = 4 and 5D, N = 2 supersymmetric Yang-Mills theories in the harmonic superspace approach. The R-symmetry transformations are constructed and their algebraic structure is studied. It is shown that such transformations are consistent both with manifest and with hidden supersymmetry transformations. These symmetries can serve as an alternative tool for constructing the relevant complete low-energy superfield effective actions determined earlier from the hidden supersymmetry considerations.