  We construct analytically stationary global configurations for both aligned
and logarithmic spiral coplanar magnetohydrodynamic (MHD) perturbations in an
axisymmetric background MHD disc with a power-law surface mass density
$\Sigma_0\propto r^{-\alpha}$, a coplanar azimuthal magnetic field $B_0\propto
r^{-\gamma}$, a consistent self-gravity and a power-law rotation curve
$v_0\propto r^{-\beta}$ where $v_0$ is the linear azimuthal gas rotation speed.
The barotropic equation of state $\Pi\propto\Sigma^{n}$ is adopted for both MHD
background equilibrium and coplanar MHD perturbations where $\Pi$ is the
vertically integrated pressure and $n$ is the barotropic index. For a
scale-free background MHD equilibrium, a relation exists among $\alpha$,
$\beta$, $\gamma$ and $n$ such that only one parameter (e.g., $\beta$) is
independent. For a linear axisymmetric stability analysis, we provide global
criteria in various parameter regimes. For nonaxisymmetric aligned and
logarithmic spiral cases, two branches of perturbation modes (i.e., fast and
slow MHD density waves) can be derived once $\beta$ is specified. To complement
the magnetized singular isothermal disc (MSID) analysis of Lou, we extend the
analysis to a wider range of $-1/4<\beta<1/2$. As an example of illustration,
we discuss specifically the $\beta=1/4$ case when the background magnetic field
is force-free. Angular momentum conservation for coplanar MHD perturbations and
other relevant aspects of our approach are discussed.
