  The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids
subject to a magnetic field and is derived via a coupling of the Navier-Stokes
equations and Maxwell's equations. Recently it has become common to study
generalizations of fluids-based differential equations. Here we consider the
generalized Magneto-Hydrodynamic alpha (gMHD-$\alpha$) system, which differs
from the original MHD system by including an additional non-linear terms
(indexed by $\alpha$), and replacing the Laplace operators by more general
Fourier multipliers with symbols of the form $-|\xi|^\gamma / g(|\xi|)$. In a
paper by Pennington, the problem was considered with initial data in the
Sobolev space $H^{s,2}(\mathbb{R}^n)$ with $n \geq 3$. Here we consider the
problem with initial data in $H^{s,p}(\mathbb{R}^n)$ with $n \geq 3$ and $p >
2$. Our goal is to minimize the regularity required for obtaining uniqueness of
a solution.
