  A new formalism for the nonlinear Alfv\'enic states sustainable in Hall
Magnetohydrodynamics is developed in a complete basis provided by the
circularly polarized Beltrami Vectors, the eigenstates of linear HMHD.
Nonlinear HMHD is, then, reduced to a rather simple looking set of scalar
equations from which a model problem of three interacting Beltrami modes is
formulated and analytically solved. The triplet interactions span a variety of
familiar nonlinear processes leading to a redistribution as well as periodic
exchange of energy. The energy exchange processes (whose strength is measured
by an energy exchange /depletion time) will, perhaps, play a dominant role in
determining the spectral content of an eventual Alfv\'enic state. All
nonlinearities (sensitive functions of the interacting wave vectors) operate at
par, and none is dominant over any substantial region of k-space; their
intricate interplay prevents a "universal" picture from emerging; few
generalizations on the processes that may, for instance, lead to a turbulent
state, are possible. However, the theory can definitely claim: 1) the energy
tends to flow from lower to higher $k$, and 2) the higher $k_z$ (in the
direction of the ambient magnetic field) components of a mode with a given $k$
are depleted/oscillate faster -in some cases much faster. It is noteworthy that
the mode coupling is the strongest (with the shortest depletion time) when the
participating wave vectors are nearly perpendicular; perhaps, an expected
consequence of the curl (cross product) nonlinearities. Numerical simulations
will be necessary to help create a fully reliable picture.
