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The role of magnetic helicity is investigated in kinetic Alfv\'en wave andoblique whistler turbulence in presence of a relatively intense externalmagnetic field $b_0 {\bf e_\parallel}$. In this situation, turbulence isstrongly anisotropic and the fluid equations describing both regimes are thereduced electron magnetohydrodynamics (REMHD) whose derivation, originally madefrom the gyrokinetic theory, is also obtained here from compressible Hall MHD.We use the asymptotic equations derived by Galtier \& Bhattacharjee (2003) tostudy the REMHD dynamics in the weak turbulence regime. The analysis is focusedon the magnetic helicity equation for which we obtain the exact solutions: theycorrespond to the entanglement relation, $n+\tilde n = -6$, where $n$ and$\tilde n$ are the power law indices of the perpendicular (to ${\bf b_0}$) wavenumber magnetic energy and helicity spectra respectively. Therefore, thespectra derived in the past from the energy equation only, namely $n=-2.5$ and$\tilde n = - 3.5$, are not the unique solutions to this problem but rathercharacterize the direct energy cascade. The solution $\tilde n = -3$ is a limitimposed by the locality condition; it is also the constant helicity fluxsolution obtained heuristically. The results obtained offer a new paradigm tounderstand solar wind turbulence at sub-ion scales where it is often observedthat $-3 < n < -2.5$.

Entanglement of helicity and energy in kinetic Alfven wave/whistler turbulence